| 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 |
| 12.2-a1 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{52} \cdot 3^{2} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( -\frac{4395631034341}{3145728} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -342 a - 677\) , \( 6146 a + 1975\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(-342a-677\right){x}+6146a+1975$ |
| 12.2-a2 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3 \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( \frac{1073716146673271}{3298534883328} a + \frac{933287514135199}{1649267441664} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -18 a - 61\) , \( 84 a - 35\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(-18a-61\right){x}+84a-35$ |
| 12.2-a3 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3^{13} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( -\frac{1073716146673271}{3298534883328} a + \frac{2940291174943669}{3298534883328} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( 160 a - 681\) , \( 2798 a + 287\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2+\left(160a-681\right){x}+2798a+287$ |
| 12.2-a4 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{20} \cdot 3^{10} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( \frac{5735339}{3888} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( 3 a + 13\) , \( -4 a - 5\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(3a+13\right){x}-4a-5$ |
| 12.2-a5 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{16} \cdot 3^{20} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.736550864$ |
1.112282735 |
\( \frac{476379541}{236196} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -17 a - 27\) , \( -20 a + 27\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(-17a-27\right){x}-20a+27$ |
| 12.2-a6 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{17} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( \frac{572452561}{6912} a + \frac{201107101}{6912} \) |
\( \bigl[1\) , \( -1\) , \( a\) , \( -20 a + 84\) , \( -55 a - 298\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3-{x}^2+\left(-20a+84\right){x}-55a-298$ |
| 12.2-a7 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{5} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( -\frac{572452561}{6912} a + \frac{386779831}{3456} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 2 a + 4\) , \( 4\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(2a+4\right){x}+4$ |
| 12.2-a8 |
12.2-a |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{32} \cdot 3^{4} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$0.347310172$ |
1.112282735 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a\) , \( 1\) , \( a\) , \( -5462 a - 10917\) , \( 389122 a + 96183\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+{x}^2+\left(-5462a-10917\right){x}+389122a+96183$ |
| 12.2-b1 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{52} \cdot 3^{2} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( -\frac{4395631034341}{3145728} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 340 a - 1019\) , \( -6147 a + 8121\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(340a-1019\right){x}-6147a+8121$ |
| 12.2-b2 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3^{13} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( \frac{1073716146673271}{3298534883328} a + \frac{933287514135199}{1649267441664} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -161 a - 521\) , \( -2799 a + 3085\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(-161a-521\right){x}-2799a+3085$ |
| 12.2-b3 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{50} \cdot 3 \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \cdot 5 \) |
$1$ |
$0.694620345$ |
1.112282735 |
\( -\frac{1073716146673271}{3298534883328} a + \frac{2940291174943669}{3298534883328} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 18 a - 79\) , \( -84 a + 49\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+\left(18a-79\right){x}-84a+49$ |
| 12.2-b4 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{20} \cdot 3^{10} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( \frac{5735339}{3888} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( -5 a + 16\) , \( 3 a - 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(-5a+16\right){x}+3a-9$ |
| 12.2-b5 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{16} \cdot 3^{20} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.736550864$ |
1.112282735 |
\( \frac{476379541}{236196} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 15 a - 44\) , \( 19 a + 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(15a-44\right){x}+19a+7$ |
| 12.2-b6 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{5} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{4} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( \frac{572452561}{6912} a + \frac{201107101}{6912} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -2 a + 6\) , \( 4\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+\left(-2a+6\right){x}+4$ |
| 12.2-b7 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{10} \cdot 3^{17} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{2} \) |
$1$ |
$3.473101729$ |
1.112282735 |
\( -\frac{572452561}{6912} a + \frac{386779831}{3456} \) |
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 19 a + 64\) , \( 54 a - 353\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3-{x}^2+\left(19a+64\right){x}+54a-353$ |
| 12.2-b8 |
12.2-b |
$8$ |
$20$ |
\(\Q(\sqrt{-39}) \) |
$2$ |
$[0, 1]$ |
12.2 |
\( 2^{2} \cdot 3 \) |
\( 2^{32} \cdot 3^{4} \) |
$1.03864$ |
$(2,a), (2,a+1), (3,a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 5$ |
2Cs, 5B |
$1$ |
\( 2^{3} \cdot 5 \) |
$1$ |
$0.347310172$ |
1.112282735 |
\( \frac{18013780041269221}{9216} \) |
\( \bigl[a + 1\) , \( -a + 1\) , \( a + 1\) , \( 5460 a - 16379\) , \( -389123 a + 485305\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3+\left(-a+1\right){x}^2+\left(5460a-16379\right){x}-389123a+485305$ |
*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.
