| 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 |
| 67600.9-CMb1 |
67600.9-CMb |
$1$ |
$1$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{12} \cdot 5^{6} \cdot 13^{9} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{yes}$ |
$-4$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$13$ |
13Cs.4.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$0.449098830$ |
0.449098830 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -174 i - 157\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-174i-157\right){x}$ |
| 67600.9-CMa1 |
67600.9-CMa |
$1$ |
$1$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{12} \cdot 5^{3} \cdot 13^{6} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{yes}$ |
$-4$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$5$ |
5Cs.4.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$1.275176390$ |
1.275176390 |
\( 1728 \) |
\( \bigl[0\) , \( 0\) , \( 0\) , \( -22 i + 19\) , \( 0\bigr] \) |
${y}^2={x}^{3}+\left(-22i+19\right){x}$ |
| 67600.9-a1 |
67600.9-a |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 5^{7} \cdot 13^{9} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.623798259$ |
0.623798259 |
\( -\frac{256}{5} a + \frac{8048}{5} \) |
\( \bigl[i + 1\) , \( 0\) , \( 0\) , \( 93 i - 74\) , \( 48 i + 112\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(93i-74\right){x}+48i+112$ |
| 67600.9-a2 |
67600.9-a |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 5^{8} \cdot 13^{9} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$0.311899129$ |
0.623798259 |
\( \frac{4439964}{25} a + \frac{3656648}{25} \) |
\( \bigl[i + 1\) , \( 0\) , \( 0\) , \( 878 i - 944\) , \( -16162 i + 7582\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(878i-944\right){x}-16162i+7582$ |
| 67600.9-b1 |
67600.9-b |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 5^{3} \cdot 13^{7} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.299396318$ |
$1.314767969$ |
3.149093514 |
\( \frac{2224}{13} a + \frac{356}{13} \) |
\( \bigl[i + 1\) , \( -i + 1\) , \( 0\) , \( 12 i - 4\) , \( 54 i - 28\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(-i+1\right){x}^{2}+\left(12i-4\right){x}+54i-28$ |
| 67600.9-b2 |
67600.9-b |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{10} \cdot 5^{3} \cdot 13^{8} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.598792636$ |
$0.657383984$ |
3.149093514 |
\( -\frac{47776420}{169} a + \frac{17266442}{169} \) |
\( \bigl[i + 1\) , \( -i + 1\) , \( 0\) , \( 232 i - 194\) , \( 2070 i - 540\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(-i+1\right){x}^{2}+\left(232i-194\right){x}+2070i-540$ |
| 67600.9-c1 |
67600.9-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{10} \cdot 5^{9} \cdot 13^{10} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.153739610$ |
0.614958440 |
\( -\frac{10359522503116}{3570125} a - \frac{4364617727362}{3570125} \) |
\( \bigl[i + 1\) , \( 0\) , \( i + 1\) , \( -3292 i - 7061\) , \( 159838 i + 210616\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-3292i-7061\right){x}+159838i+210616$ |
| 67600.9-c2 |
67600.9-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{10} \cdot 5^{18} \cdot 13^{7} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.153739610$ |
0.614958440 |
\( \frac{2896194844812}{3173828125} a + \frac{3398200522034}{3173828125} \) |
\( \bigl[i + 1\) , \( 0\) , \( i + 1\) , \( 1468 i + 1119\) , \( 14542 i + 22988\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(1468i+1119\right){x}+14542i+22988$ |
| 67600.9-c3 |
67600.9-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 5^{12} \cdot 13^{8} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2Cs |
$1$ |
\( 2^{5} \) |
$1$ |
$0.307479220$ |
0.614958440 |
\( -\frac{3643553424}{2640625} a + \frac{4710369332}{2640625} \) |
\( \bigl[i + 1\) , \( 0\) , \( i + 1\) , \( -272 i - 451\) , \( 1258 i + 3426\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-272i-451\right){x}+1258i+3426$ |
| 67600.9-c4 |
67600.9-c |
$4$ |
$4$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{4} \cdot 5^{9} \cdot 13^{7} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$0.614958440$ |
0.614958440 |
\( \frac{50931328}{1625} a + \frac{11807696}{1625} \) |
\( \bigl[i + 1\) , \( 0\) , \( i + 1\) , \( -192 i - 136\) , \( -1388 i - 127\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(-192i-136\right){x}-1388i-127$ |
| 67600.9-d1 |
67600.9-d |
$1$ |
$1$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{8} \cdot 5^{6} \cdot 13^{2} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$2$ |
2Cn |
$1$ |
\( 1 \) |
$1$ |
$2.533498384$ |
2.533498384 |
\( 512 a + 768 \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -2 i + 6\) , \( 6 i - 5\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-2i+6\right){x}+6i-5$ |
| 67600.9-e1 |
67600.9-e |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{12} \cdot 5^{9} \cdot 13^{9} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Cn |
$1$ |
\( 2^{5} \) |
$1$ |
$0.302143887$ |
2.417151103 |
\( \frac{906876}{2197} a + \frac{1118799}{2197} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -30 i - 379\) , \( 3497 i + 1526\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-30i-379\right){x}+3497i+1526$ |
| 67600.9-e2 |
67600.9-e |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{12} \cdot 5^{9} \cdot 13^{12} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 3$ |
2B, 3Cn |
$4$ |
\( 2^{4} \) |
$1$ |
$0.151071943$ |
2.417151103 |
\( -\frac{10047446145}{4826809} a + \frac{17756992962}{4826809} \) |
\( \bigl[i + 1\) , \( i\) , \( i + 1\) , \( -650 i + 2461\) , \( 36109 i + 21142\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+i{x}^{2}+\left(-650i+2461\right){x}+36109i+21142$ |
| 67600.9-f1 |
67600.9-f |
$4$ |
$10$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{13} \cdot 5^{16} \cdot 13^{3} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.359365980$ |
2.874927842 |
\( -\frac{80398914857}{19531250} a - \frac{197826917099}{19531250} \) |
\( \bigl[i + 1\) , \( i + 1\) , \( i + 1\) , \( -309 i - 452\) , \( -4378 i - 3383\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(i+1\right){x}^{2}+\left(-309i-452\right){x}-4378i-3383$ |
| 67600.9-f2 |
67600.9-f |
$4$ |
$10$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{14} \cdot 5^{11} \cdot 13^{3} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.718731960$ |
2.874927842 |
\( \frac{10462207}{6250} a - \frac{2706038}{3125} \) |
\( \bigl[i + 1\) , \( i + 1\) , \( i + 1\) , \( -89 i + 8\) , \( 130 i - 339\bigr] \) |
${y}^2+\left(i+1\right){x}{y}+\left(i+1\right){y}={x}^{3}+\left(i+1\right){x}^{2}+\left(-89i+8\right){x}+130i-339$ |
| 67600.9-f3 |
67600.9-f |
$4$ |
$10$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{22} \cdot 5^{7} \cdot 13^{3} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$0.718731960$ |
2.874927842 |
\( -\frac{523313}{160} a + \frac{424661}{40} \) |
\( \bigl[i + 1\) , \( 0\) , \( 0\) , \( 61 i + 125\) , \( 484 i - 365\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(61i+125\right){x}+484i-365$ |
| 67600.9-f4 |
67600.9-f |
$4$ |
$10$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{17} \cdot 5^{8} \cdot 13^{3} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2, 5$ |
2B, 5B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.359365980$ |
2.874927842 |
\( -\frac{12916359143}{200} a + \frac{17274394699}{200} \) |
\( \bigl[i + 1\) , \( 0\) , \( 0\) , \( 941 i + 1965\) , \( 31012 i - 24461\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(941i+1965\right){x}+31012i-24461$ |
| 67600.9-g1 |
67600.9-g |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{10} \cdot 5^{9} \cdot 13^{7} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$0.518674966$ |
4.149399729 |
\( -\frac{109298}{1625} a + \frac{1963264}{1625} \) |
\( \bigl[i + 1\) , \( i + 1\) , \( 0\) , \( -132 i - 84\) , \( -104 i - 460\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(-132i-84\right){x}-104i-460$ |
| 67600.9-g2 |
67600.9-g |
$2$ |
$2$ |
\(\Q(\sqrt{-1}) \) |
$2$ |
$[0, 1]$ |
67600.9 |
\( 2^{4} \cdot 5^{2} \cdot 13^{2} \) |
\( 2^{11} \cdot 5^{12} \cdot 13^{8} \) |
$2.88174$ |
$(a+1), (2a+1), (2a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$2$ |
2B |
$1$ |
\( 2^{6} \) |
$1$ |
$0.259337483$ |
4.149399729 |
\( \frac{321047281}{2640625} a + \frac{6395175767}{2640625} \) |
\( \bigl[i + 1\) , \( i + 1\) , \( 0\) , \( 658 i + 386\) , \( 196 i - 4560\bigr] \) |
${y}^2+\left(i+1\right){x}{y}={x}^{3}+\left(i+1\right){x}^{2}+\left(658i+386\right){x}+196i-4560$ |
*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.
