Here we do some sample verifications of full BSD.

1 14A1

1. t = 6 (Cremona)

2. We need 2, 7 to split in K, with D prime to 14. D = −1, −2 no good. D = −3 works.

3. For e=ellinit([1,0,1,4,-6]), ellrootno(e)= 1, so we use E^{D}.

4. reducing E^{D}–[126, [1, 0, 1/2, 1/2], 36] ellanalyticrank(e2) INPUT CURVE NOT MINIMAL MINIMISED TO [1, -1, 1, 40, 155] This also has analytic rank 0, so E/K has rank 0, and our choice of D was bad, so we go back to step

2. D = −19 gives the next K satisfying the Heegner hypothesis. Compute E^{−19}:

SO STILL RANK ZERO.

We have to try higher D’s. try −31. new twist is [0, −31/4, 0, 31^{2} ∗ 9/2, 23 ∗ 31^{3}/4]. This finally has rank 1, so −31 is okay, but it has conductor 13454. But now this isn’t listed in Cremona’s table (he lists a different rep of the isogeny class, apparently), so I can’t find a M-W generator.

Magma gives free generator (2635 : −203732 : 1) of E^{−}31. Height is 3.65369257693788807017...

5. α = \sqrt{31}/ω_{E}. L(E/Q, 1) = 0.3302239655664549236801327451, L′(E^{D}/Q, 1) = 5.218903558405753668020265818. ω_{E} ∼ 2.626251405582016186497854965. Then

ĥ(y_{K}) ∼ \sqrt{31}∗0.3302239655664549236801327451∗5.218903558405753668020265818/2.626251405582016186497854965 ∼ 3.653695993510223589610801579

.

6. I_{K} ∼ 3.653695993510223589610801579/3.65369257693788807017 ∼ 1.000000935101206129067749992, which is good news. (didn’t sqrt)

7. � (E/Q)[p] = 0 for all p ≠ 2, 3.

8. Perform 2 and 3-descent to show triviality of � there as well.

2 15A1

1. t = 2.

2. D = −11 is first to satisfy Heegner hypothesis.

3. root number is 1, so we need the twist E^{−}11 e2 = ellinit([0, −55/4, 0, −19 ∗ 11^{2}/2, 39∗11^{3}/4]); ellanalyticrank(e2)INP UT CURV ENOT MINIMALMINIMISEDT O[1, 1, 0, −1212, 7011]Summing85a_{n}termsRankisoddL^{(}1) = 3.20742659864952933563593124736 = [1, 3.207426598649529335635931247, 1.666073986584487822978044210]

4. Has conductor 1815, and a M-W generator is (1527 : 38880 : 1) according to Magma. Height is 1.66607395687729953337.

5. L^{1}(E^{D}/Q, 1) ∼ 3.207426598649529335635931247, L(E/Q, 1) ∼ 0.3501506898191986108956287229.

ω_{E} ∼ 2.235701712617529101766742778, so

ĥ(y_{K}) ∼ 1.666073649878152404321456746

6. I_{K} ∼ .9999998157349823146788753820 (didn’t bother w/ sqrt)

7. � [p] = 0 for p ≠ 2.

8. Run 2-descent. So twice the problem of twists with high conductor has arisen. We’ll now try

an example where E/Q has rank 1 to avoid this problem.

3 37A1

1. t = 2

2. D = −4 is ok (remember, have to use discriminant of K.)

3. root number is −1, so we can compute with E.

4. a M-W generator is (x, y) = (0, 0).

5. height of Heegner point. 0.3059997890560586325057899275 = L′(E/Q, 1) the twist by −1 is isomorphic to [0, 0, 0, −1, −1/4] 2.451389679618052965620183312 = L(E^{−1}/Q, 1). u = 2, \sqrt{|D|} = 2, ω_{E} = 7.338132740789576739070720999. then α = 0.5450977982131191944086112918, and finally

ĥ(y_{K}) = .4088913359049488054582505759 ∗ 2

6. ĥ(0, 0) = 0.05111140823996884023588609975 (twice the pari value), so I_{K} = \sqrt{2 ∗ 8.000001369267654590724456920}

7. Conclude that � (E/Q)[p] = 0 for p ≠ 2.

8. Perform 2-descent.

4 17A1

1. t = 2

2. D = −4 works.

3. E = ([1, −1, 1, −1, −14]) root number is 1, so the twist has rank 1. A W.E.

for the twist is y^{2} = x^{3} − 14256 ∗ x + 41523840, and this has conductor 272, so we can look it up in Cremona’s tables. Using MAGMA to find a minimal model, the twist is isomorphic to 272B4. Cremona only gives gen’s for strong Weil curves, so we use MAGMA again to find z = (−36 : 6480 : 1). For future reference we find the height to be 2.036297434690738317186.

4. L(E/Q, 1) = 0.3867705802219959913534634548, L′(E^{D}/Q, 1) = 1.397785337254319520109796667, ω_{E} = 2.123938699064983080555718069. Then

ĥ(y_{K}) ∼ 2.036300750689832872531797350

5. I_{K} ∼ 1.000000814222344949138440895.

6. � (E/Q)[p] = 0 for all p ≠ 2.

7. perform 2-descent.

Now we will try a couple of high-conductor examples, one for each possible root number.

5 462F1

1. t = 2

2. 462 = 2 ∗ 3 ∗ 7 ∗ 11. D = −17 works.

3. E = ([1, 0, 0, −97, 1337]). root number is 1, so we need to find generator of E^{D}. E^{D} has a W.E. y^{2} = x^{3} −36338571∗x−308321515206, and its conductor is 1068144. Magma’s not going to find a generator here, so we’re stuck for the moment. Perhaps trying higher values of |D| we might luck out and get a lower conductor twist? Try D = −227 just for kicks. 23806398 is the new conductor, even worse. There probably just aren’t enough curves of low conductor... My experimentation in the next section allows me to return here with a twist of relatively low conductor, letting D = −841 (conductor 3696). But unfortunately this has rank 0, so we’re still out of luck: E/Q(\sqrt{−841} has rank 0.

6 462E1

1. t = 2

2. We can take D = −17 as above.

3. E = ([1, 1, 1, −405, 4731]). root number is −1, so we need the free generator of E.

4. a M-W generator is (−17, 92) from Cremona’s tables. its height is ∼ .039411059881925059045.

5. u = 1, L(E^{D}/Q, 1) ∼ L′(E/Q, 1) ∼ 2.036386747116844897014687695, ω_{E} ∼ 0.3161575749688195692943622398. Both MAGMA and PARI say that E^{D} also has rank 1, so the original E/K must have had rank 2. Now again we’ll go back and try D = −227. Again I find that the twist also has rank 1, so E has rank 2 over Q(\sqrt{−227}) and Q(\sqrt{−17}). Yet another value of D must be tried. D = −229 works as well (N.B. I’ve gotten all my values of D by applying Q.R. and solving a system of congruences). Again, the twist has rank 1. Contrast this with the situation with 14A1, where we needed to try a few values of D before getting E/K to have rank > 0. Now we’re having trouble getting this rank < 2. Try −17 − 462 = −479. I’m getting that the twist E^{−479} has rank 2 over Q!. Now I try D = −841, and get rank zero, with L(E^{−841}/Q, 1) ∼ 1.027947342001393896663733734. A minimal model, incidentally, is [0, 1, 0, −6480, −315756]. We can find the height of a Heegner point here:

ĥ(y_{K}) ∼ 192.0107464816242787400060455

6. I_{K} ∼ 69.79972486951426762605949488. This is not convincingly close to an integer... how precise are all of the above calculations? Or did I make an error along the way?

7. ?

8. ?

http://magma.maths.usyd.edu.au/magma/ps/hb7.ps is a handy reference for everything about ell. curves and magma