| 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 |
| 9.2-a1 |
9.2-a |
$2$ |
$2$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{8} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2^{2} \) |
$1$ |
$128.2657412$ |
3.826823960 |
\( -\frac{57552988}{81} a^{3} + \frac{203474506}{81} a^{2} + \frac{360460997}{81} a - \frac{1022161409}{81} \) |
\( \bigl[-\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 15\) , \( -\frac{1}{2} a^{3} + \frac{3}{2} a^{2} + \frac{7}{2} a - 7\) , \( -a^{3} + 4 a^{2} + 7 a - 22\) , \( 3 a^{3} - 13 a^{2} - 15 a + 68\) , \( 13 a^{3} - 48 a^{2} - 77 a + 241\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a^{2}+\frac{5}{2}a-15\right){x}{y}+\left(-a^{3}+4a^{2}+7a-22\right){y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{3}{2}a^{2}+\frac{7}{2}a-7\right){x}^{2}+\left(3a^{3}-13a^{2}-15a+68\right){x}+13a^{3}-48a^{2}-77a+241$ |
| 9.2-a2 |
9.2-a |
$2$ |
$2$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{4} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$0 \le r \le 1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
|
\( 2 \) |
$1$ |
$128.2657412$ |
3.826823960 |
\( \frac{120936947786}{9} a^{3} - \frac{403435113823}{9} a^{2} - \frac{871665540226}{9} a + \frac{2278009429337}{9} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{7}{2} a + 9\) , \( -\frac{1}{2} a^{3} + \frac{3}{2} a^{2} + \frac{5}{2} a - 7\) , \( 1\) , \( \frac{65}{2} a^{3} + \frac{81}{2} a^{2} - \frac{899}{2} a - 864\) , \( \frac{971}{2} a^{3} + \frac{913}{2} a^{2} - \frac{12981}{2} a - 11486\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{7}{2}a+9\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{2}a^{3}+\frac{3}{2}a^{2}+\frac{5}{2}a-7\right){x}^{2}+\left(\frac{65}{2}a^{3}+\frac{81}{2}a^{2}-\frac{899}{2}a-864\right){x}+\frac{971}{2}a^{3}+\frac{913}{2}a^{2}-\frac{12981}{2}a-11486$ |
| 9.2-b1 |
9.2-b |
$8$ |
$12$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{6} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.198693060$ |
$921.7020045$ |
3.962427342 |
\( \frac{75722293}{18} a^{3} - \frac{781238521}{54} a^{2} - \frac{534563591}{18} a + \frac{2256140557}{27} \) |
\( \bigl[-a^{3} + 4 a^{2} + 7 a - 23\) , \( -a^{3} + 4 a^{2} + 5 a - 24\) , \( 0\) , \( \frac{3}{2} a^{3} - \frac{1}{2} a^{2} - \frac{59}{2} a - 40\) , \( 714 a^{3} + 2139 a^{2} - 2226 a - 7767\bigr] \) |
${y}^2+\left(-a^{3}+4a^{2}+7a-23\right){x}{y}={x}^{3}+\left(-a^{3}+4a^{2}+5a-24\right){x}^{2}+\left(\frac{3}{2}a^{3}-\frac{1}{2}a^{2}-\frac{59}{2}a-40\right){x}+714a^{3}+2139a^{2}-2226a-7767$ |
| 9.2-b2 |
9.2-b |
$8$ |
$12$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{2} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$2.384316726$ |
$921.7020045$ |
3.962427342 |
\( \frac{2132236814813}{6} a^{3} - \frac{7112953729471}{6} a^{2} - \frac{15368320399639}{6} a + \frac{20081756499046}{3} \) |
\( \bigl[-\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 14\) , \( \frac{1}{2} a^{3} - \frac{5}{2} a^{2} - \frac{5}{2} a + 15\) , \( 1\) , \( -\frac{3}{2} a^{3} - \frac{13}{2} a^{2} + \frac{15}{2} a + 34\) , \( \frac{7}{2} a^{3} + \frac{19}{2} a^{2} - \frac{33}{2} a - 46\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a^{2}+\frac{5}{2}a-14\right){x}{y}+{y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a^{2}-\frac{5}{2}a+15\right){x}^{2}+\left(-\frac{3}{2}a^{3}-\frac{13}{2}a^{2}+\frac{15}{2}a+34\right){x}+\frac{7}{2}a^{3}+\frac{19}{2}a^{2}-\frac{33}{2}a-46$ |
| 9.2-b3 |
9.2-b |
$8$ |
$12$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{12} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.397386121$ |
$921.7020045$ |
3.962427342 |
\( \frac{1128335}{1458} a^{3} + \frac{2633299}{1458} a^{2} - \frac{4905073}{1458} a - \frac{1857094}{729} \) |
\( \bigl[-\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{7}{2} a - 14\) , \( -a + 1\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 15\) , \( \frac{3}{2} a^{3} - \frac{5}{2} a^{2} - \frac{21}{2} a + 14\) , \( 2 a^{3} - 4 a^{2} - 13 a + 20\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a^{2}+\frac{7}{2}a-14\right){x}{y}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a^{2}+\frac{5}{2}a-15\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(\frac{3}{2}a^{3}-\frac{5}{2}a^{2}-\frac{21}{2}a+14\right){x}+2a^{3}-4a^{2}-13a+20$ |
| 9.2-b4 |
9.2-b |
$8$ |
$12$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{24} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \cdot 3 \) |
$0.794772242$ |
$57.60637528$ |
3.962427342 |
\( -\frac{3773323479509}{1062882} a^{3} + \frac{4417801365773}{354294} a^{2} + \frac{23285202760375}{1062882} a - \frac{11003098731317}{177147} \) |
\( \bigl[-\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{7}{2} a - 14\) , \( -a + 1\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 15\) , \( 9 a^{3} - 30 a^{2} - 58 a + 149\) , \( \frac{59}{2} a^{3} - \frac{225}{2} a^{2} - \frac{377}{2} a + 553\bigr] \) |
${y}^2+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a^{2}+\frac{7}{2}a-14\right){x}{y}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a^{2}+\frac{5}{2}a-15\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(9a^{3}-30a^{2}-58a+149\right){x}+\frac{59}{2}a^{3}-\frac{225}{2}a^{2}-\frac{377}{2}a+553$ |
| 9.2-b5 |
9.2-b |
$8$ |
$12$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{6} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 3 \) |
$0.794772242$ |
$921.7020045$ |
3.962427342 |
\( \frac{760447277}{18} a^{3} + \frac{6738087007}{54} a^{2} - \frac{2525059723}{18} a - \frac{12691427689}{27} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{7}{2} a + 8\) , \( -\frac{3}{2} a^{3} + \frac{11}{2} a^{2} + \frac{21}{2} a - 32\) , \( -\frac{1}{2} a^{3} + \frac{5}{2} a^{2} + \frac{5}{2} a - 14\) , \( -\frac{217}{2} a^{3} + \frac{763}{2} a^{2} + \frac{1335}{2} a - 1896\) , \( 1600 a^{3} - 5621 a^{2} - 9873 a + 28008\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{7}{2}a+8\right){x}{y}+\left(-\frac{1}{2}a^{3}+\frac{5}{2}a^{2}+\frac{5}{2}a-14\right){y}={x}^{3}+\left(-\frac{3}{2}a^{3}+\frac{11}{2}a^{2}+\frac{21}{2}a-32\right){x}^{2}+\left(-\frac{217}{2}a^{3}+\frac{763}{2}a^{2}+\frac{1335}{2}a-1896\right){x}+1600a^{3}-5621a^{2}-9873a+28008$ |
| 9.2-b6 |
9.2-b |
$8$ |
$12$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{2} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 1 \) |
$0.596079181$ |
$921.7020045$ |
3.962427342 |
\( -\frac{2459967279581}{6} a^{3} + \frac{8642915486659}{6} a^{2} + \frac{15176171944507}{6} a - \frac{21532067517496}{3} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{7}{2} a + 8\) , \( a^{3} - 4 a^{2} - 5 a + 23\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{7}{2} a + 9\) , \( -\frac{5}{2} a^{3} + \frac{17}{2} a^{2} + \frac{33}{2} a - 42\) , \( a^{3} - 3 a^{2} - 10 a + 4\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{7}{2}a+8\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{7}{2}a+9\right){y}={x}^{3}+\left(a^{3}-4a^{2}-5a+23\right){x}^{2}+\left(-\frac{5}{2}a^{3}+\frac{17}{2}a^{2}+\frac{33}{2}a-42\right){x}+a^{3}-3a^{2}-10a+4$ |
| 9.2-b7 |
9.2-b |
$8$ |
$12$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( 3^{4} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2Cs, 3B |
$1$ |
\( 2 \) |
$1.192158363$ |
$921.7020045$ |
3.962427342 |
\( -\frac{2959175}{18} a^{3} - \frac{931219}{18} a^{2} + \frac{33013633}{18} a + \frac{33908785}{9} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + 9\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + 8\) , \( -a^{3} + 4 a^{2} + 6 a - 22\) , \( -\frac{163}{2} a^{3} + \frac{555}{2} a^{2} + \frac{1183}{2} a - 1558\) , \( 641 a^{3} - 2126 a^{2} - 4613 a + 12014\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{5}{2}a+9\right){x}{y}+\left(-a^{3}+4a^{2}+6a-22\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{5}{2}a+8\right){x}^{2}+\left(-\frac{163}{2}a^{3}+\frac{555}{2}a^{2}+\frac{1183}{2}a-1558\right){x}+641a^{3}-2126a^{2}-4613a+12014$ |
| 9.2-b8 |
9.2-b |
$8$ |
$12$ |
4.4.19225.1 |
$4$ |
$[4, 0]$ |
9.2 |
\( 3^{2} \) |
\( - 3^{8} \) |
$16.30617$ |
$(3/2a^3-11/2a^2-19/2a+31)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$2.384316726$ |
$57.60637528$ |
3.962427342 |
\( -\frac{148735529594573}{162} a^{3} - \frac{44389227275927}{54} a^{2} + \frac{1978654649544127}{162} a + \frac{575471171779328}{27} \) |
\( \bigl[\frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + 9\) , \( \frac{1}{2} a^{3} - \frac{3}{2} a^{2} - \frac{5}{2} a + 8\) , \( -a^{3} + 4 a^{2} + 6 a - 22\) , \( -64 a^{3} + 220 a^{2} + 464 a - 1238\) , \( \frac{1809}{2} a^{3} - \frac{6007}{2} a^{2} - \frac{13031}{2} a + 16959\bigr] \) |
${y}^2+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{5}{2}a+9\right){x}{y}+\left(-a^{3}+4a^{2}+6a-22\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{3}{2}a^{2}-\frac{5}{2}a+8\right){x}^{2}+\left(-64a^{3}+220a^{2}+464a-1238\right){x}+\frac{1809}{2}a^{3}-\frac{6007}{2}a^{2}-\frac{13031}{2}a+16959$ |
*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.
