| 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 |
| 4.2-a1 |
4.2-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{16} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$321.6765974$ |
1.674581477 |
\( -\frac{16203}{1024} a^{3} + \frac{83967}{256} a^{2} + \frac{154837}{512} a - \frac{1902139}{1024} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{23}{4}\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( \frac{1}{4} a^{3} + 3 a^{2} + \frac{3}{2} a - \frac{47}{4}\) , \( -\frac{7}{4} a^{3} + 2 a^{2} + \frac{35}{2} a - \frac{7}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{23}{4}\right){x}^{2}+\left(\frac{1}{4}a^{3}+3a^{2}+\frac{3}{2}a-\frac{47}{4}\right){x}-\frac{7}{4}a^{3}+2a^{2}+\frac{35}{2}a-\frac{7}{4}$ |
| 4.2-a2 |
4.2-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{2} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$643.3531949$ |
1.674581477 |
\( \frac{1231316059}{8} a^{3} + 279826331 a^{2} - \frac{3002163331}{4} a - \frac{8301356289}{8} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( -\frac{1}{4} a^{3} + \frac{5}{2} a + \frac{3}{4}\) , \( a + 1\) , \( -\frac{13}{4} a^{3} - 5 a^{2} + \frac{37}{2} a + \frac{95}{4}\) , \( -\frac{55}{4} a^{3} - 25 a^{2} + \frac{135}{2} a + \frac{373}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{5}{2}a+\frac{3}{4}\right){x}^{2}+\left(-\frac{13}{4}a^{3}-5a^{2}+\frac{37}{2}a+\frac{95}{4}\right){x}-\frac{55}{4}a^{3}-25a^{2}+\frac{135}{2}a+\frac{373}{4}$ |
| 4.2-a3 |
4.2-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{4} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$321.6765974$ |
1.674581477 |
\( \frac{26230390783955}{16} a^{3} - \frac{14427521023955}{4} a^{2} - \frac{96524160204805}{8} a + \frac{415294904348323}{16} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a + \frac{5}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{15}{4}\) , \( \frac{7}{4} a^{3} - 9 a^{2} + \frac{25}{2} a - \frac{13}{4}\) , \( \frac{23}{4} a^{3} - 32 a^{2} + \frac{99}{2} a - \frac{65}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{15}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{5}{4}\right){x}^{2}+\left(\frac{7}{4}a^{3}-9a^{2}+\frac{25}{2}a-\frac{13}{4}\right){x}+\frac{23}{4}a^{3}-32a^{2}+\frac{99}{2}a-\frac{65}{4}$ |
| 4.2-a4 |
4.2-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1286.706389$ |
1.674581477 |
\( \frac{18319255}{64} a^{3} - 633055 a^{2} - \frac{67502955}{32} a + \frac{292378267}{64} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a + \frac{5}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{15}{4}\) , \( \frac{7}{4} a^{3} - 4 a^{2} - \frac{5}{2} a + \frac{27}{4}\) , \( -\frac{9}{4} a^{3} + 11 a^{2} - \frac{3}{2} a - \frac{97}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{15}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{5}{4}\right){x}^{2}+\left(\frac{7}{4}a^{3}-4a^{2}-\frac{5}{2}a+\frac{27}{4}\right){x}-\frac{9}{4}a^{3}+11a^{2}-\frac{3}{2}a-\frac{97}{4}$ |
| 4.2-a5 |
4.2-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{4} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1286.706389$ |
1.674581477 |
\( -\frac{152221843}{16} a^{3} + \frac{140841667}{4} a^{2} + \frac{987237}{8} a - \frac{1072029027}{16} \) |
\( \bigl[a^{2} - 4\) , \( -a^{2} - a + 6\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{5}{4}\) , \( -\frac{3}{4} a^{3} + \frac{5}{2} a - \frac{3}{4}\) , \( -\frac{15}{4} a^{3} + 7 a^{2} + \frac{53}{2} a - \frac{215}{4}\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{5}{4}\right){y}={x}^{3}+\left(-a^{2}-a+6\right){x}^{2}+\left(-\frac{3}{4}a^{3}+\frac{5}{2}a-\frac{3}{4}\right){x}-\frac{15}{4}a^{3}+7a^{2}+\frac{53}{2}a-\frac{215}{4}$ |
| 4.2-a6 |
4.2-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{2} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$160.8382987$ |
1.674581477 |
\( -\frac{28652036771020955}{8} a^{3} + 13249114371850221 a^{2} + \frac{206040815108515}{4} a - \frac{201676595279423167}{8} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{15}{4}\) , \( -\frac{1}{4} a^{3} - a^{2} + \frac{3}{2} a + \frac{15}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{15}{4}\) , \( 23 a^{3} - 46 a^{2} - 167 a + 334\) , \( -\frac{307}{2} a^{3} + 351 a^{2} + 1135 a - \frac{5009}{2}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{15}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{15}{4}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}-a^{2}+\frac{3}{2}a+\frac{15}{4}\right){x}^{2}+\left(23a^{3}-46a^{2}-167a+334\right){x}-\frac{307}{2}a^{3}+351a^{2}+1135a-\frac{5009}{2}$ |
| 4.2-b1 |
4.2-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{4} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$283.9821496$ |
1.478352020 |
\( \frac{26230390783955}{16} a^{3} - \frac{14427521023955}{4} a^{2} - \frac{96524160204805}{8} a + \frac{415294904348323}{16} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{15}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{15}{4}\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( -69 a^{3} + 193 a^{2} + 546 a - 1356\) , \( 1285 a^{3} - 2709 a^{2} - 9407 a + 19721\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{15}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{15}{4}\right){x}^{2}+\left(-69a^{3}+193a^{2}+546a-1356\right){x}+1285a^{3}-2709a^{2}-9407a+19721$ |
| 4.2-b2 |
4.2-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{2} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$1$ |
$567.9642993$ |
1.478352020 |
\( \frac{1231316059}{8} a^{3} + 279826331 a^{2} - \frac{3002163331}{4} a - \frac{8301356289}{8} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{15}{4}\) , \( -\frac{1}{4} a^{3} - a^{2} + \frac{1}{2} a + \frac{15}{4}\) , \( 1\) , \( -\frac{11}{4} a^{3} - 2 a^{2} + \frac{51}{2} a + \frac{117}{4}\) , \( -\frac{53}{4} a^{3} - 12 a^{2} + \frac{215}{2} a + \frac{467}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{15}{4}\right){x}{y}+{y}={x}^{3}+\left(-\frac{1}{4}a^{3}-a^{2}+\frac{1}{2}a+\frac{15}{4}\right){x}^{2}+\left(-\frac{11}{4}a^{3}-2a^{2}+\frac{51}{2}a+\frac{117}{4}\right){x}-\frac{53}{4}a^{3}-12a^{2}+\frac{215}{2}a+\frac{467}{4}$ |
| 4.2-b3 |
4.2-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{4} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1135.928598$ |
1.478352020 |
\( -\frac{152221843}{16} a^{3} + \frac{140841667}{4} a^{2} + \frac{987237}{8} a - \frac{1072029027}{16} \) |
\( \bigl[a\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{5}{2} a - \frac{15}{4}\) , \( 0\) , \( -\frac{1}{2} a^{3} + a^{2} + a - \frac{3}{2}\) , \( 0\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{5}{2}a-\frac{15}{4}\right){x}^{2}+\left(-\frac{1}{2}a^{3}+a^{2}+a-\frac{3}{2}\right){x}$ |
| 4.2-b4 |
4.2-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{2} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 1 \) |
$1$ |
$141.9910748$ |
1.478352020 |
\( -\frac{28652036771020955}{8} a^{3} + 13249114371850221 a^{2} + \frac{206040815108515}{4} a - \frac{201676595279423167}{8} \) |
\( \bigl[a\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{5}{2} a - \frac{15}{4}\) , \( 0\) , \( 2 a^{3} - 4 a^{2} - 4 a + 6\) , \( \frac{23}{2} a^{3} - 31 a^{2} - 10 a + \frac{107}{2}\bigr] \) |
${y}^2+a{x}{y}={x}^{3}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{5}{2}a-\frac{15}{4}\right){x}^{2}+\left(2a^{3}-4a^{2}-4a+6\right){x}+\frac{23}{2}a^{3}-31a^{2}-10a+\frac{107}{2}$ |
| 4.2-b5 |
4.2-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{8} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1$ |
$1135.928598$ |
1.478352020 |
\( \frac{18319255}{64} a^{3} - 633055 a^{2} - \frac{67502955}{32} a + \frac{292378267}{64} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}\) , \( -\frac{1}{4} a^{3} + \frac{5}{2} a + \frac{3}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{19}{4}\) , \( -\frac{141}{4} a^{3} + 76 a^{2} + \frac{521}{2} a - \frac{2193}{4}\) , \( 375 a^{3} - 826 a^{2} - 2759 a + 5942\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{19}{4}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{5}{2}a+\frac{3}{4}\right){x}^{2}+\left(-\frac{141}{4}a^{3}+76a^{2}+\frac{521}{2}a-\frac{2193}{4}\right){x}+375a^{3}-826a^{2}-2759a+5942$ |
| 4.2-b6 |
4.2-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
4.2 |
\( 2^{2} \) |
\( 2^{16} \) |
$10.20656$ |
$(1/2a^3+a^2-2a-7/2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$283.9821496$ |
1.478352020 |
\( -\frac{16203}{1024} a^{3} + \frac{83967}{256} a^{2} + \frac{154837}{512} a - \frac{1902139}{1024} \) |
\( \bigl[1\) , \( \frac{1}{4} a^{3} - a^{2} - \frac{3}{2} a + \frac{21}{4}\) , \( a\) , \( \frac{1}{4} a^{3} - a^{2} - \frac{5}{2} a + \frac{37}{4}\) , \( \frac{5}{4} a^{3} - \frac{19}{2} a - \frac{15}{4}\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(\frac{1}{4}a^{3}-a^{2}-\frac{3}{2}a+\frac{21}{4}\right){x}^{2}+\left(\frac{1}{4}a^{3}-a^{2}-\frac{5}{2}a+\frac{37}{4}\right){x}+\frac{5}{4}a^{3}-\frac{19}{2}a-\frac{15}{4}$ |
*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.
