| 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 |
| 115.1-A1 |
115.1-A |
$8$ |
$12$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
115.1 |
\( 5 \cdot 23 \) |
\( - 5^{2} \cdot 23 \) |
$0.94504$ |
$(a^2+1), (-a^2-2a+2)$ |
0 |
$\Z/12\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$67.03413881$ |
0.388266227 |
\( -\frac{296772}{575} a^{2} + \frac{501473}{575} a - \frac{384109}{575} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( a\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+a{x}$ |
| 115.1-A2 |
115.1-A |
$8$ |
$12$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
115.1 |
\( 5 \cdot 23 \) |
\( 5^{4} \cdot 23^{2} \) |
$0.94504$ |
$(a^2+1), (-a^2-2a+2)$ |
0 |
$\Z/2\Z\oplus\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$33.51706940$ |
0.388266227 |
\( \frac{524530283647}{330625} a^{2} - \frac{936669243273}{330625} a + \frac{716408628984}{330625} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( 6 a - 5\) , \( 4 a^{2} - 7 a + 2\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(6a-5\right){x}+4a^{2}-7a+2$ |
| 115.1-A3 |
115.1-A |
$8$ |
$12$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
115.1 |
\( 5 \cdot 23 \) |
\( - 5^{6} \cdot 23^{3} \) |
$0.94504$ |
$(a^2+1), (-a^2-2a+2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$7.448237646$ |
0.388266227 |
\( -\frac{258468891972347}{190109375} a^{2} - \frac{300129881653427}{190109375} a - \frac{79092285157884}{190109375} \) |
\( \bigl[a^{2}\) , \( -a^{2} - 1\) , \( 0\) , \( 6 a^{2} + 15 a + 10\) , \( 29 a^{2} - 28 a - 39\bigr] \) |
${y}^2+a^{2}{x}{y}={x}^{3}+\left(-a^{2}-1\right){x}^{2}+\left(6a^{2}+15a+10\right){x}+29a^{2}-28a-39$ |
| 115.1-A4 |
115.1-A |
$8$ |
$12$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
115.1 |
\( 5 \cdot 23 \) |
\( 5^{2} \cdot 23^{4} \) |
$0.94504$ |
$(a^2+1), (-a^2-2a+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2^{2} \) |
$1$ |
$8.379267352$ |
0.388266227 |
\( -\frac{5581242271849501}{6996025} a^{2} - \frac{4930264753084216}{6996025} a - \frac{541308225756447}{6996025} \) |
\( \bigl[a^{2} + 1\) , \( a + 1\) , \( a + 1\) , \( 80 a^{2} + 75 a + 10\) , \( -199 a^{2} + 536 a + 518\bigr] \) |
${y}^2+\left(a^{2}+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(80a^{2}+75a+10\right){x}-199a^{2}+536a+518$ |
| 115.1-A5 |
115.1-A |
$8$ |
$12$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
115.1 |
\( 5 \cdot 23 \) |
\( 5^{8} \cdot 23 \) |
$0.94504$ |
$(a^2+1), (-a^2-2a+2)$ |
0 |
$\Z/6\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$16.75853470$ |
0.388266227 |
\( \frac{75716587374138607891}{8984375} a^{2} - \frac{132873348136199426744}{8984375} a + \frac{100303122964106562977}{8984375} \) |
\( \bigl[a + 1\) , \( a^{2} + a\) , \( 0\) , \( -1347 a^{2} + 2368 a - 1788\) , \( 34299 a^{2} - 60188 a + 45434\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}+a\right){x}^{2}+\left(-1347a^{2}+2368a-1788\right){x}+34299a^{2}-60188a+45434$ |
| 115.1-A6 |
115.1-A |
$8$ |
$12$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
115.1 |
\( 5 \cdot 23 \) |
\( 5^{12} \cdot 23^{6} \) |
$0.94504$ |
$(a^2+1), (-a^2-2a+2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$3.724118823$ |
0.388266227 |
\( -\frac{24639055700669505303}{36141574462890625} a^{2} - \frac{34086830170754179598}{36141574462890625} a + \frac{95255415238700540209}{36141574462890625} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -15 a^{2} + 6 a - 10\) , \( -31 a^{2} + 6 a - 8\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-15a^{2}+6a-10\right){x}-31a^{2}+6a-8$ |
| 115.1-A7 |
115.1-A |
$8$ |
$12$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
115.1 |
\( 5 \cdot 23 \) |
\( 5^{6} \cdot 23^{12} \) |
$0.94504$ |
$(a^2+1), (-a^2-2a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2^{2} \) |
$1$ |
$0.931029705$ |
0.388266227 |
\( \frac{477008324446884679504423036}{342416006750317515625} a^{2} - \frac{836616303539449005035716299}{342416006750317515625} a + \frac{631813749335688590262064917}{342416006750317515625} \) |
\( \bigl[1\) , \( -a^{2} + 1\) , \( a + 1\) , \( -341 a^{2} + 550 a - 401\) , \( -3926 a^{2} + 6899 a - 5146\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+1\right){x}^{2}+\left(-341a^{2}+550a-401\right){x}-3926a^{2}+6899a-5146$ |
| 115.1-A8 |
115.1-A |
$8$ |
$12$ |
3.1.23.1 |
$3$ |
$[1, 1]$ |
115.1 |
\( 5 \cdot 23 \) |
\( 5^{24} \cdot 23^{3} \) |
$0.94504$ |
$(a^2+1), (-a^2-2a+2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.862059411$ |
0.388266227 |
\( \frac{20007944569370849968678844}{725209712982177734375} a^{2} - \frac{7918974152713453255537371}{725209712982177734375} a + \frac{4956482510946229795232693}{725209712982177734375} \) |
\( \bigl[a\) , \( a + 1\) , \( a\) , \( -130 a^{2} + 61 a - 30\) , \( 520 a^{2} - 496 a - 4\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-130a^{2}+61a-30\right){x}+520a^{2}-496a-4$ |
*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.
