| 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 |
| 72.1-a1 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$2.170705473$ |
$2.325279868$ |
4.209904133 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -942 a + 4522\) , \( 248884 a - 1193604\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-942a+4522\right){x}+248884a-1193604$ |
| 72.1-a2 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$4.341410946$ |
$18.60223895$ |
4.209904133 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+{x}$ |
| 72.1-a3 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$8.682821893$ |
$37.20447790$ |
4.209904133 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 258 a - 1233\) , \( -3006 a + 14418\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(258a-1233\right){x}-3006a+14418$ |
| 72.1-a4 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$4.341410946$ |
$9.301119475$ |
4.209904133 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 1458 a - 6988\) , \( 67524 a - 323832\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(1458a-6988\right){x}+67524a-323832$ |
| 72.1-a5 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$4.341410946$ |
$37.20447790$ |
4.209904133 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( 3858 a - 18498\) , \( -275046 a + 1319076\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(3858a-18498\right){x}-275046a+1319076$ |
| 72.1-a6 |
72.1-a |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{4} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$8.682821893$ |
$2.325279868$ |
4.209904133 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -23058 a - 110578\) , \( -4238844 a - 20328780\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-23058a-110578\right){x}-4238844a-20328780$ |
| 72.1-b1 |
72.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{4} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$12.26725338$ |
2.557899152 |
\( \frac{4}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -2 a + 14\) , \( -2 a + 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-2a+14\right){x}-2a+12$ |
| 72.1-b2 |
72.1-b |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{8} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$12.26725338$ |
2.557899152 |
\( \frac{3370318}{81} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -2 a + 4\) , \( 2 a - 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-2a+4\right){x}+2a-8$ |
| 72.1-c1 |
72.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{20} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.504017794$ |
1.305309507 |
\( -\frac{14647977776}{59049} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -2 a - 67\) , \( 70 a - 150\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-2a-67\right){x}+70a-150$ |
| 72.1-c2 |
72.1-c |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{10} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.504017794$ |
1.305309507 |
\( \frac{15043017316604}{243} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -2 a - 1282\) , \( 3958 a - 2580\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-2a-1282\right){x}+3958a-2580$ |
| 72.1-d1 |
72.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{12} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.410556525$ |
2.317811777 |
\( -\frac{21296}{729} \) |
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( a + 1\) , \( 2 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}+2a+4$ |
| 72.1-d2 |
72.1-d |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{6} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$14.82111305$ |
2.317811777 |
\( \frac{4499456}{27} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( -2080 a - 9975\) , \( 111800 a + 536174\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+\left(-2080a-9975\right){x}+111800a+536174$ |
| 72.1-e1 |
72.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{12} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.410556525$ |
2.317811777 |
\( -\frac{21296}{729} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -3 a + 1\) , \( -3 a + 4\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-3a+1\right){x}-3a+4$ |
| 72.1-e2 |
72.1-e |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{6} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$14.82111305$ |
2.317811777 |
\( \frac{4499456}{27} \) |
\( \bigl[0\) , \( -1\) , \( 0\) , \( -2080 a - 9975\) , \( -111800 a - 536174\bigr] \) |
${y}^2={x}^{3}-{x}^{2}+\left(-2080a-9975\right){x}-111800a-536174$ |
| 72.1-f1 |
72.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{20} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.504017794$ |
1.305309507 |
\( -\frac{14647977776}{59049} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 2 a - 67\) , \( -70 a - 150\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(2a-67\right){x}-70a-150$ |
| 72.1-f2 |
72.1-f |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{10} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$2.504017794$ |
1.305309507 |
\( \frac{15043017316604}{243} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 2 a - 1282\) , \( -3958 a - 2580\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(2a-1282\right){x}-3958a-2580$ |
| 72.1-g1 |
72.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{4} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$12.26725338$ |
2.557899152 |
\( \frac{4}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 2 a + 14\) , \( 2 a + 12\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(2a+14\right){x}+2a+12$ |
| 72.1-g2 |
72.1-g |
$2$ |
$2$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{8} \) |
$2.49670$ |
$(-a+5), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$12.26725338$ |
2.557899152 |
\( \frac{3370318}{81} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 2 a + 4\) , \( -2 a - 8\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(2a+4\right){x}-2a-8$ |
| 72.1-h1 |
72.1-h |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{16} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$12.14060112$ |
$5.683508517$ |
3.596936714 |
\( \frac{207646}{6561} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( -938 a + 4522\) , \( -252644 a + 1211660\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-938a+4522\right){x}-252644a+1211660$ |
| 72.1-h2 |
72.1-h |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1.517575141$ |
$11.36701703$ |
3.596936714 |
\( \frac{2048}{3} \) |
\( \bigl[0\) , \( 1\) , \( 0\) , \( 1\) , \( 0\bigr] \) |
${y}^2={x}^{3}+{x}^{2}+{x}$ |
| 72.1-h3 |
72.1-h |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{4} \cdot 3^{4} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$3.035150282$ |
$22.73403407$ |
3.596936714 |
\( \frac{35152}{9} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 262 a - 1233\) , \( 4046 a - 19382\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(262a-1233\right){x}+4046a-19382$ |
| 72.1-h4 |
72.1-h |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{8} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2Cs |
$1$ |
\( 2^{3} \) |
$6.070300564$ |
$22.73403407$ |
3.596936714 |
\( \frac{1556068}{81} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 1462 a - 6988\) , \( -61684 a + 295848\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(1462a-6988\right){x}-61684a+295848$ |
| 72.1-h5 |
72.1-h |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{8} \cdot 3^{2} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2 \) |
$6.070300564$ |
$5.683508517$ |
3.596936714 |
\( \frac{28756228}{3} \) |
\( \bigl[a + 1\) , \( 0\) , \( 0\) , \( 3862 a - 18498\) , \( 290486 a - 1393100\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(3862a-18498\right){x}+290486a-1393100$ |
| 72.1-h6 |
72.1-h |
$6$ |
$8$ |
\(\Q(\sqrt{23}) \) |
$2$ |
$[2, 0]$ |
72.1 |
\( 2^{3} \cdot 3^{2} \) |
\( 2^{10} \cdot 3^{4} \) |
$2.49670$ |
$(-a+5), (3)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
|
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$3.035150282$ |
$22.73403407$ |
3.596936714 |
\( \frac{3065617154}{9} \) |
\( \bigl[a + 1\) , \( -a\) , \( 0\) , \( -23062 a - 110578\) , \( 4146604 a + 19886436\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}-a{x}^{2}+\left(-23062a-110578\right){x}+4146604a+19886436$ |
*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.
