| 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 |
| 128772.4-a1 |
128772.4-a |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{2} \cdot 3^{27} \cdot 7^{5} \cdot 73 \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$1$ |
$0.307617628$ |
1.420824965 |
\( -\frac{2603394015538}{4435583733} a + \frac{39798652173097}{8871167466} \) |
\( \bigl[1\) , \( a\) , \( 0\) , \( 333 a - 711\) , \( -3753 a + 4914\bigr] \) |
${y}^2+{x}{y}={x}^{3}+a{x}^{2}+\left(333a-711\right){x}-3753a+4914$ |
| 128772.4-b1 |
128772.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{8} \cdot 73^{4} \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.455345452$ |
2.103150558 |
\( -\frac{2913552868954332}{3341024655409} a - \frac{3278482882753907}{6682049310818} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -115 a - 111\) , \( 961 a + 823\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-115a-111\right){x}+961a+823$ |
| 128772.4-b2 |
128772.4-b |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{4} \cdot 3^{6} \cdot 7^{4} \cdot 73^{2} \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.910690905$ |
2.103150558 |
\( \frac{5321822571516}{1827847} a + \frac{2824357089119}{7311388} \) |
\( \bigl[1\) , \( a\) , \( 1\) , \( -115 a - 141\) , \( 853 a + 523\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-115a-141\right){x}+853a+523$ |
| 128772.4-c1 |
128772.4-c |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{2} \cdot 3^{7} \cdot 7^{5} \cdot 73 \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{3} \) |
$0.135886582$ |
$1.728467666$ |
4.339375048 |
\( -\frac{21138360659}{1051638} a - \frac{1968536650}{525819} \) |
\( \bigl[1\) , \( -a + 1\) , \( a + 1\) , \( -15 a - 18\) , \( 44 a + 13\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-15a-18\right){x}+44a+13$ |
| 128772.4-d1 |
128772.4-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{2} \cdot 3^{12} \cdot 7^{8} \cdot 73^{2} \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.387502995$ |
1.789799669 |
\( -\frac{4171535580289717}{33855382134} a - \frac{6435309105271451}{33855382134} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -873 a + 844\) , \( -150 a + 9669\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-873a+844\right){x}-150a+9669$ |
| 128772.4-d2 |
128772.4-d |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{4} \cdot 3^{18} \cdot 7^{4} \cdot 73 \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.775005990$ |
1.789799669 |
\( \frac{15158438249}{73013724} a + \frac{2390251168}{6084477} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -63 a + 34\) , \( 174 a + 111\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-63a+34\right){x}+174a+111$ |
| 128772.4-e1 |
128772.4-e |
$1$ |
$1$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{2} \cdot 3^{9} \cdot 7^{3} \cdot 73 \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
|
|
$1$ |
\( 2^{2} \) |
$0.274089680$ |
$1.954803978$ |
4.949430776 |
\( \frac{1961142413}{64386} a - \frac{79820281}{64386} \) |
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -19 a - 7\) , \( -39 a + 9\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-19a-7\right){x}-39a+9$ |
| 128772.4-f1 |
128772.4-f |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{20} \cdot 3^{6} \cdot 7^{6} \cdot 73 \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \cdot 5 \) |
$0.092682141$ |
$0.620362817$ |
5.311304161 |
\( \frac{1089514891873}{179479552} a - \frac{438743146785}{179479552} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -114 a - 65\) , \( 846 a - 75\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-114a-65\right){x}+846a-75$ |
| 128772.4-f2 |
128772.4-f |
$2$ |
$2$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{10} \cdot 3^{6} \cdot 7^{12} \cdot 73^{2} \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{7} \cdot 5 \) |
$0.046341070$ |
$0.310181408$ |
5.311304161 |
\( -\frac{804510685664741}{983059984928} a + \frac{4274401951807}{30720624529} \) |
\( \bigl[a\) , \( -1\) , \( a\) , \( -114 a + 415\) , \( 3630 a + 1461\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-114a+415\right){x}+3630a+1461$ |
| 128772.4-g1 |
128772.4-g |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{12} \cdot 3^{6} \cdot 7^{4} \cdot 73^{3} \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1[2] |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1$ |
$0.577765893$ |
2.668586355 |
\( \frac{8487617049675}{152494664} a - \frac{14473502427339}{1219957312} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -104 a - 224\) , \( 942 a + 1178\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(-104a-224\right){x}+942a+1178$ |
| 128772.4-g2 |
128772.4-g |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{2} \cdot 3^{6} \cdot 7^{15} \cdot 73^{2} \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1[2] |
$1$ |
\( 2^{5} \cdot 3^{2} \) |
$1$ |
$0.288882946$ |
2.668586355 |
\( \frac{637509743623817073}{147520438988258} a - \frac{217278319585716315}{73760219494129} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -168 a + 735\) , \( 8379 a - 3087\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-168a+735\right){x}+8379a-3087$ |
| 128772.4-g3 |
128772.4-g |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{4} \cdot 3^{6} \cdot 7^{12} \cdot 73 \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1[2] |
$1$ |
\( 2^{4} \cdot 3^{2} \) |
$1$ |
$0.577765893$ |
2.668586355 |
\( -\frac{275361962643}{34353508} a + \frac{19029031083}{34353508} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -198 a + 195\) , \( 141 a - 1137\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-198a+195\right){x}+141a-1137$ |
| 128772.4-g4 |
128772.4-g |
$4$ |
$6$ |
\(\Q(\sqrt{-3}) \) |
$2$ |
$[0, 1]$ |
128772.4 |
\( 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 73 \) |
\( 2^{6} \cdot 3^{6} \cdot 7^{5} \cdot 73^{6} \) |
$2.93194$ |
$(-2a+1), (-3a+1), (3a-2), (9a-8), (2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3B.1.1[2] |
$1$ |
\( 2^{5} \cdot 3^{2} \) |
$1$ |
$0.288882946$ |
2.668586355 |
\( -\frac{1454621964199278075}{1453413909279556} a + \frac{6478079565720766761}{2906827818559112} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 16 a - 584\) , \( 2142 a - 4102\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(16a-584\right){x}+2142a-4102$ |
*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.
