| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 64.7-a1 |
64.7-a |
$1$ |
$1$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 1 \) |
$1$ |
$28.55411158$ |
4.970632811 |
\( -373248 a - 884736 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( -3 a - 7\) , \( 4 a + 9\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-3a-7\right){x}+4a+9$ |
| 64.7-b1 |
64.7-b |
$1$ |
$1$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.772306970$ |
$12.50248858$ |
3.361703814 |
\( 512 a \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 46 a + 111\) , \( 59 a + 139\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(46a+111\right){x}+59a+139$ |
| 64.7-c1 |
64.7-c |
$2$ |
$3$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{14} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$12.51227705$ |
2.178107860 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( a + 3\) , \( -1\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+3\right){x}-1$ |
| 64.7-c2 |
64.7-c |
$2$ |
$3$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{14} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-3$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$12.51227705$ |
2.178107860 |
\( 0 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( a + 3\) , \( -7335 a + 24734\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+3\right){x}-7335a+24734$ |
| 64.7-d1 |
64.7-d |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( - 2^{15} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$1$ |
$19.04890146$ |
1.657994057 |
\( -729 a + 2457 \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -3 a - 8\) , \( 149 a + 353\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-3a-8\right){x}+149a+353$ |
| 64.7-d2 |
64.7-d |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 1 \) |
$1$ |
$38.09780292$ |
1.657994057 |
\( -47877075 a + 161457003 \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a - 7\) , \( -2 a + 3\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-7\right){x}-2a+3$ |
| 64.7-d3 |
64.7-d |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{12} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cs, 3Nn |
$1$ |
\( 2^{2} \) |
$1$ |
$38.09780292$ |
1.657994057 |
\( 10935 a + 43281 \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 78 a - 265\) , \( -593 a + 1999\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(78a-265\right){x}-593a+1999$ |
| 64.7-d4 |
64.7-d |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( - 2^{15} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2 \) |
$1$ |
$19.04890146$ |
1.657994057 |
\( 1097736219 a + 2604139173 \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -57 a + 190\) , \( -2485 a + 8379\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-57a+190\right){x}-2485a+8379$ |
| 64.7-e1 |
64.7-e |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.255540541$ |
$18.02937231$ |
1.604033535 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( -706 a + 2381\) , \( -6626 a + 22340\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-706a+2381\right){x}-6626a+22340$ |
| 64.7-e2 |
64.7-e |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.810945957$ |
$1.639033846$ |
1.604033535 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -61 a - 139\) , \( -386 a - 913\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-61a-139\right){x}-386a-913$ |
| 64.7-e3 |
64.7-e |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs |
$1$ |
\( 1 \) |
$0.085180180$ |
$54.08811693$ |
1.604033535 |
\( -32768 \) |
\( \bigl[0\) , \( -1\) , \( a + 1\) , \( 154 a - 519\) , \( -1861 a + 6271\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(154a-519\right){x}-1861a+6271$ |
| 64.7-e4 |
64.7-e |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-11$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$3$ |
3Cs |
$1$ |
\( 1 \) |
$0.936981985$ |
$4.917101539$ |
1.604033535 |
\( -32768 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -a + 1\) , \( -a - 2\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+1\right){x}-a-2$ |
| 64.7-e5 |
64.7-e |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.255540541$ |
$18.02937231$ |
1.604033535 |
\( 6548115718144 a - 22082088337408 \) |
\( \bigl[0\) , \( a\) , \( a + 1\) , \( 7167 a + 17004\) , \( 229955 a + 545517\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(7167a+17004\right){x}+229955a+545517$ |
| 64.7-e6 |
64.7-e |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.810945957$ |
$1.639033846$ |
1.604033535 |
\( 6548115718144 a - 22082088337408 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 9 a - 9\) , \( 14 a - 81\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(9a-9\right){x}+14a-81$ |
| 64.7-f1 |
64.7-f |
$1$ |
$1$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3Nn |
$1$ |
\( 1 \) |
$1$ |
$7.828642438$ |
1.362791725 |
\( -373248 a - 884736 \) |
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( -43 a + 145\) , \( -311 a + 1044\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-43a+145\right){x}-311a+1044$ |
| 64.7-g1 |
64.7-g |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( - 2^{15} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \) |
$1$ |
$13.72025503$ |
0.597097458 |
\( -729 a + 2457 \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 7 a - 16\) , \( -8 a + 31\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(7a-16\right){x}-8a+31$ |
| 64.7-g2 |
64.7-g |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 1 \) |
$1$ |
$13.72025503$ |
0.597097458 |
\( -47877075 a + 161457003 \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -618 a - 1466\) , \( 4553 a + 10801\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-618a-1466\right){x}+4553a+10801$ |
| 64.7-g3 |
64.7-g |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{12} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2Cs, 3Nn |
$1$ |
\( 2^{2} \) |
$1$ |
$13.72025503$ |
0.597097458 |
\( 10935 a + 43281 \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -5 a - 12\) , \( -24 a - 57\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-5a-12\right){x}-24a-57$ |
| 64.7-g4 |
64.7-g |
$4$ |
$4$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( - 2^{15} \) |
$1.45191$ |
$(-a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2, 3$ |
2B, 3Nn |
$1$ |
\( 2^{2} \) |
$1$ |
$3.430063758$ |
0.597097458 |
\( 1097736219 a + 2604139173 \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -100 a - 237\) , \( -1101 a - 2612\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-100a-237\right){x}-1101a-2612$ |
| 64.7-h1 |
64.7-h |
$1$ |
$1$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.7 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a+3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.102899136$ |
$29.73410704$ |
1.065220847 |
\( 512 a \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( 2\) , \( -a - 2\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+2{x}-a-2$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
