| 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 |
| 25.1-a1 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.961531894$ |
1.082695022 |
\( -\frac{359104782699}{244140625} a - \frac{52148361654}{48828125} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -26 a - 43\) , \( 84 a + 148\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-26a-43\right){x}+84a+148$ |
| 25.1-a2 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.961531894$ |
1.082695022 |
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 29 a - 66\) , \( -125 a + 369\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(29a-66\right){x}-125a+369$ |
| 25.1-a3 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.84612757$ |
1.082695022 |
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 29 a - 71\) , \( -118 a + 338\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(29a-71\right){x}-118a+338$ |
| 25.1-a4 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$19.84612757$ |
1.082695022 |
\( -\frac{22825881}{125} a + \frac{12909294}{25} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -a - 3\) , \( -a - 2\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-a-3\right){x}-a-2$ |
| 25.1-a5 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$19.84612757$ |
1.082695022 |
\( -\frac{32714515537919631}{125} a + \frac{91315629670496661}{125} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 429 a - 1196\) , \( -7403 a + 20663\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(429a-1196\right){x}-7403a+20663$ |
| 25.1-a6 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$19.84612757$ |
1.082695022 |
\( \frac{22825881}{125} a + \frac{41720589}{125} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 4 a - 1\) , \( 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-1\right){x}+9$ |
| 25.1-a7 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \) |
$1$ |
$19.84612757$ |
1.082695022 |
\( \frac{118077162021}{15625} a + \frac{42307984341}{3125} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -26 a - 48\) , \( 72 a + 129\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-26a-48\right){x}+72a+129$ |
| 25.1-a8 |
25.1-a |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$1$ |
$19.84612757$ |
1.082695022 |
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \) |
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -426 a - 773\) , \( 6632 a + 11894\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-426a-773\right){x}+6632a+11894$ |
| 25.1-b1 |
25.1-b |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.949771685$ |
1.276425190 |
\( -\frac{359104782699}{244140625} a - \frac{52148361654}{48828125} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( -18 a + 58\) , \( -20 a + 51\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-18a+58\right){x}-20a+51$ |
| 25.1-b2 |
25.1-b |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{4} \cdot 3 \) |
$1$ |
$1.949771685$ |
1.276425190 |
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( 22 a + 33\) , \( 56 a + 96\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(22a+33\right){x}+56a+96$ |
| 25.1-b3 |
25.1-b |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.799086741$ |
1.276425190 |
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -3 a - 12\) , \( -4 a - 11\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-3a-12\right){x}-4a-11$ |
| 25.1-b4 |
25.1-b |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$31.19634696$ |
1.276425190 |
\( -\frac{22825881}{125} a + \frac{12909294}{25} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 7 a - 7\) , \( -5 a + 26\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7a-7\right){x}-5a+26$ |
| 25.1-b5 |
25.1-b |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 3 \) |
$1$ |
$1.949771685$ |
1.276425190 |
\( -\frac{32714515537919631}{125} a + \frac{91315629670496661}{125} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -28 a - 137\) , \( -284 a - 786\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-28a-137\right){x}-284a-786$ |
| 25.1-b6 |
25.1-b |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$1$ |
$31.19634696$ |
1.276425190 |
\( \frac{22825881}{125} a + \frac{41720589}{125} \) |
\( \bigl[a\) , \( a\) , \( a\) , \( -3 a - 7\) , \( a + 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-3a-7\right){x}+a+1$ |
| 25.1-b7 |
25.1-b |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$7.799086741$ |
1.276425190 |
\( \frac{118077162021}{15625} a + \frac{42307984341}{3125} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 7 a - 12\) , \( -5 a + 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(7a-12\right){x}-5a+9$ |
| 25.1-b8 |
25.1-b |
$8$ |
$12$ |
\(\Q(\sqrt{21}) \) |
$2$ |
$[2, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$0.91566$ |
$(-a), (-a+1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
✓ |
|
$2, 3$ |
2B, 3B |
$4$ |
\( 3 \) |
$1$ |
$1.949771685$ |
1.276425190 |
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \) |
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 32 a - 162\) , \( 150 a - 921\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(32a-162\right){x}+150a-921$ |
*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.
