| 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 |
| 1.1-a1 |
1.1-a |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.51333$ |
$\textsf{none}$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.2 |
$1$ |
\( 1 \) |
$1$ |
$2.562583394$ |
0.446088510 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a\) , \( 1\) , \( -435 a - 1030\) , \( -7890 a - 18717\bigr] \) |
${y}^2+{y}={x}^{3}-a{x}^{2}+\left(-435a-1030\right){x}-7890a-18717$ |
| 1.1-a2 |
1.1-a |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
1.1 |
\( 1 \) |
\( 1 \) |
$0.51333$ |
$\textsf{none}$ |
0 |
$\Z/3\Z$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
✓ |
✓ |
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$23.06325055$ |
0.446088510 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( a - 1\) , \( 1\) , \( -25 a + 85\) , \( 72 a - 243\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-25a+85\right){x}+72a-243$ |
| 64.6-e1 |
64.6-e |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$2.810945957$ |
$1.639033846$ |
1.604033535 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( a\) , \( a\) , \( -9 a\) , \( -15 a - 66\bigr] \) |
${y}^2+a{y}={x}^{3}+a{x}^{2}-9a{x}-15a-66$ |
| 64.6-e2 |
64.6-e |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
64.6 |
\( 2^{6} \) |
\( 2^{6} \) |
$1.45191$ |
$(-a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.255540541$ |
$18.02937231$ |
1.604033535 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a + 1\) , \( a\) , \( -7167 a + 24171\) , \( -229956 a + 775473\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-7167a+24171\right){x}-229956a+775473$ |
| 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 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a + 1\) , \( a + 1\) , \( -61 a - 139\) , \( -386 a - 913\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-61a-139\right){x}-386a-913$ |
| 121.1-b1 |
121.1-b |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{6} \) |
$1.70252$ |
$(-4a-9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11B.1.3 |
$1$ |
\( 2 \) |
$0.269355468$ |
$7.687750184$ |
1.441876556 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -110 a - 227\) , \( 1012 a + 2419\bigr] \) |
${y}^2+{y}={x}^{3}-{x}^{2}+\left(-110a-227\right){x}+1012a+2419$ |
| 121.1-b2 |
121.1-b |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
121.1 |
\( 11^{2} \) |
\( 11^{6} \) |
$1.70252$ |
$(-4a-9)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
$11$ |
11B.1.8 |
$1$ |
\( 2 \) |
$2.962910153$ |
$0.698886380$ |
1.441876556 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( a\) , \( 1\) , \( -12371 a + 41722\) , \( 513055 a - 1730165\bigr] \) |
${y}^2+{y}={x}^{3}+a{x}^{2}+\left(-12371a+41722\right){x}+513055a-1730165$ |
| 144.4-a1 |
144.4-a |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
144.4 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$1.77822$ |
$(-a+3), (-2a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$9$ |
\( 1 \) |
$1$ |
$0.739754106$ |
1.158971947 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( -1153 a - 2731\) , \( -34456 a - 81737\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1153a-2731\right){x}-34456a-81737$ |
| 144.4-a2 |
144.4-a |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
144.4 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$1.77822$ |
$(-a+3), (-2a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$6.657786957$ |
1.158971947 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( -1329 a + 4485\) , \( 17121 a - 57738\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1329a+4485\right){x}+17121a-57738$ |
| 144.5-a1 |
144.5-a |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
144.5 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$1.77822$ |
$(-a-2), (-2a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$9$ |
\( 1 \) |
$1$ |
$0.739754106$ |
1.158971947 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( -23659 a - 56125\) , \( -3391842 a - 8046405\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-23659a-56125\right){x}-3391842a-8046405$ |
| 144.5-a2 |
144.5-a |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
144.5 |
\( 2^{4} \cdot 3^{2} \) |
\( 2^{12} \cdot 3^{6} \) |
$1.77822$ |
$(-a-2), (-2a+7)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$6.657786957$ |
1.158971947 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( a + 1\) , \( a\) , \( -75 a + 195\) , \( 239 a - 701\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-75a+195\right){x}+239a-701$ |
| 256.1-g1 |
256.1-g |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$2.05331$ |
$(-a-2), (-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$1$ |
$5.765812638$ |
1.003699148 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( a\) , \( 0\) , \( -6965 a - 16520\) , \( 528427 a + 1253576\bigr] \) |
${y}^2={x}^{3}+a{x}^{2}+\left(-6965a-16520\right){x}+528427a+1253576$ |
| 256.1-g2 |
256.1-g |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
256.1 |
\( 2^{8} \) |
\( 2^{24} \) |
$2.05331$ |
$(-a-2), (-a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$9$ |
\( 1 \) |
$1$ |
$0.640645848$ |
1.003699148 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -395 a + 1315\) , \( -3275 a + 11027\bigr] \) |
${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-395a+1315\right){x}-3275a+11027$ |
| 576.6-j1 |
576.6-j |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
576.6 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$2.51479$ |
$(-a-2), (-2a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.873457767$ |
$10.40926295$ |
3.165446055 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( -819 a - 1941\) , \( 22847 a + 54199\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-819a-1941\right){x}+22847a+54199$ |
| 576.6-j2 |
576.6-j |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
576.6 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$2.51479$ |
$(-a-2), (-2a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$9.608035443$ |
$0.946296632$ |
3.165446055 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( -a - 1\) , \( a\) , \( -467 a + 1579\) , \( 3484 a - 11753\bigr] \) |
${y}^2+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-467a+1579\right){x}+3484a-11753$ |
| 576.7-j1 |
576.7-j |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
576.7 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$2.51479$ |
$(-a+3), (-2a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$9.608035443$ |
$0.946296632$ |
3.165446055 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( -8321 a - 19739\) , \( -712481 a - 1690206\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-8321a-19739\right){x}-712481a-1690206$ |
| 576.7-j2 |
576.7-j |
$6$ |
$99$ |
\(\Q(\sqrt{33}) \) |
$2$ |
$[2, 0]$ |
576.7 |
\( 2^{6} \cdot 3^{2} \) |
\( 2^{6} \cdot 3^{6} \) |
$2.51479$ |
$(-a+3), (-2a+7)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{potential}$ |
$-99$ |
$N(\mathrm{U}(1))$ |
✓ |
|
|
✓ |
|
|
$1$ |
\( 1 \) |
$0.873457767$ |
$10.40926295$ |
3.165446055 |
\( -6548115718144 a - 15533972619264 \) |
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( -49 a + 149\) , \( -92 a + 317\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-49a+149\right){x}-92a+317$ |
*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.
