| 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 |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.188938793$ |
$1163.230861$ |
2.288252076 |
\( \frac{506072727552}{25} a^{3} - \frac{1113403817359}{25} a^{2} - \frac{3724555649524}{25} a + \frac{8012299584804}{25} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( a^{2} + a - 5\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}\) , \( \frac{539}{4} a^{3} + 172 a^{2} - \frac{2085}{2} a - \frac{5657}{4}\) , \( -\frac{8175}{4} a^{3} - 2362 a^{2} + \frac{31729}{2} a + \frac{79337}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(\frac{539}{4}a^{3}+172a^{2}-\frac{2085}{2}a-\frac{5657}{4}\right){x}-\frac{8175}{4}a^{3}-2362a^{2}+\frac{31729}{2}a+\frac{79337}{4}$ |
| 25.1-a2 |
25.1-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.377877586$ |
$1163.230861$ |
2.288252076 |
\( \frac{157869}{5} a^{3} - \frac{328006}{5} a^{2} - \frac{1190227}{5} a + \frac{2483489}{5} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( a^{2} + a - 5\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}\) , \( \frac{27}{2} a^{3} + 17 a^{2} - 100 a - \frac{251}{2}\) , \( \frac{11}{2} a^{3} + 4 a^{2} - 34 a - \frac{31}{2}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}\right){y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(\frac{27}{2}a^{3}+17a^{2}-100a-\frac{251}{2}\right){x}+\frac{11}{2}a^{3}+4a^{2}-34a-\frac{31}{2}$ |
| 25.1-a3 |
25.1-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.755755172$ |
$290.8077153$ |
2.288252076 |
\( \frac{35308842}{5} a^{3} + \frac{64261067}{5} a^{2} - \frac{171923378}{5} a - \frac{237845696}{5} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( \frac{1}{4} a^{3} - a^{2} - \frac{3}{2} a + \frac{17}{4}\) , \( a^{2} + a - 5\) , \( -a^{3} - 2 a^{2} + 4 a + 10\) , \( -a^{3} - 3 a^{2} + 6 a + 9\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-a^{2}-\frac{3}{2}a+\frac{17}{4}\right){x}^{2}+\left(-a^{3}-2a^{2}+4a+10\right){x}-a^{3}-3a^{2}+6a+9$ |
| 25.1-a4 |
25.1-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.094469396$ |
$290.8077153$ |
2.288252076 |
\( -\frac{8951325797632}{625} a^{3} - \frac{9678146215168}{625} a^{2} + \frac{13874284280433}{125} a + \frac{81718315223174}{625} \) |
\( \bigl[a^{2} - 4\) , \( \frac{1}{4} a^{3} - \frac{5}{2} a + \frac{5}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{15}{4}\) , \( -\frac{253}{2} a^{3} + 291 a^{2} + 869 a - \frac{3871}{2}\) , \( 1946 a^{3} - 4201 a^{2} - 14741 a + 31346\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{15}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{5}{2}a+\frac{5}{4}\right){x}^{2}+\left(-\frac{253}{2}a^{3}+291a^{2}+869a-\frac{3871}{2}\right){x}+1946a^{3}-4201a^{2}-14741a+31346$ |
| 25.1-a5 |
25.1-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.377877586$ |
$290.8077153$ |
2.288252076 |
\( \frac{40501594119568875520256}{5} a^{3} - \frac{89106785247241445375744}{5} a^{2} - \frac{298080586628446171957273}{5} a + \frac{641231989324594775655826}{5} \) |
\( \bigl[a^{2} - 4\) , \( \frac{1}{4} a^{3} - a^{2} - \frac{1}{2} a + \frac{17}{4}\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}\) , \( \frac{1461}{4} a^{3} + 660 a^{2} - \frac{3583}{2} a - \frac{9847}{4}\) , \( \frac{39997}{2} a^{3} + 36382 a^{2} - 97446 a - \frac{269545}{2}\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-a^{2}-\frac{1}{2}a+\frac{17}{4}\right){x}^{2}+\left(\frac{1461}{4}a^{3}+660a^{2}-\frac{3583}{2}a-\frac{9847}{4}\right){x}+\frac{39997}{2}a^{3}+36382a^{2}-97446a-\frac{269545}{2}$ |
| 25.1-a6 |
25.1-a |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.755755172$ |
$290.8077153$ |
2.288252076 |
\( -\frac{197515092}{5} a^{3} + \frac{730667933}{5} a^{2} + \frac{2842878}{5} a - \frac{1390258114}{5} \) |
\( \bigl[1\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{15}{4}\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( -\frac{7}{4} a^{3} - 3 a^{2} + \frac{17}{2} a + \frac{41}{4}\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{15}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){x}^{2}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){x}-\frac{7}{4}a^{3}-3a^{2}+\frac{17}{2}a+\frac{41}{4}$ |
| 25.1-b1 |
25.1-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1.469838141$ |
$5.175558822$ |
2.534507336 |
\( -\frac{8951325797632}{625} a^{3} - \frac{9678146215168}{625} a^{2} + \frac{13874284280433}{125} a + \frac{81718315223174}{625} \) |
\( \bigl[a\) , \( a^{2} - 5\) , \( a^{2} - 4\) , \( -23 a^{3} + 58 a^{2} + 40 a - 128\) , \( 65 a^{3} - 367 a^{2} + 141 a + 710\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-5\right){x}^{2}+\left(-23a^{3}+58a^{2}+40a-128\right){x}+65a^{3}-367a^{2}+141a+710$ |
| 25.1-b2 |
25.1-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.939676283$ |
$1324.943058$ |
2.534507336 |
\( -\frac{197515092}{5} a^{3} + \frac{730667933}{5} a^{2} + \frac{2842878}{5} a - \frac{1390258114}{5} \) |
\( \bigl[a\) , \( a^{2} + a - 5\) , \( 1\) , \( -\frac{1}{4} a^{3} + 4 a^{2} + \frac{9}{2} a - \frac{81}{4}\) , \( 2 a^{3} - a^{2} - 12 a + 14\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a^{2}+a-5\right){x}^{2}+\left(-\frac{1}{4}a^{3}+4a^{2}+\frac{9}{2}a-\frac{81}{4}\right){x}+2a^{3}-a^{2}-12a+14$ |
| 25.1-b3 |
25.1-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$0.734919070$ |
$82.80894116$ |
2.534507336 |
\( \frac{506072727552}{25} a^{3} - \frac{1113403817359}{25} a^{2} - \frac{3724555649524}{25} a + \frac{8012299584804}{25} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a - \frac{5}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( 6 a^{3} + 4 a^{2} - 59 a - 68\) , \( \frac{121}{4} a^{3} + 29 a^{2} - \frac{501}{2} a - \frac{1171}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a-\frac{5}{4}\right){x}^{2}+\left(6a^{3}+4a^{2}-59a-68\right){x}+\frac{121}{4}a^{3}+29a^{2}-\frac{501}{2}a-\frac{1171}{4}$ |
| 25.1-b4 |
25.1-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.469838141$ |
$1324.943058$ |
2.534507336 |
\( \frac{157869}{5} a^{3} - \frac{328006}{5} a^{2} - \frac{1190227}{5} a + \frac{2483489}{5} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a - \frac{5}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( -\frac{1}{4} a^{3} - a^{2} - \frac{3}{2} a + \frac{3}{4}\) , \( 3 a^{3} + 6 a^{2} - 13 a - 20\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a-\frac{5}{4}\right){x}^{2}+\left(-\frac{1}{4}a^{3}-a^{2}-\frac{3}{2}a+\frac{3}{4}\right){x}+3a^{3}+6a^{2}-13a-20$ |
| 25.1-b5 |
25.1-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.734919070$ |
$1324.943058$ |
2.534507336 |
\( \frac{35308842}{5} a^{3} + \frac{64261067}{5} a^{2} - \frac{171923378}{5} a - \frac{237845696}{5} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}\) , \( -\frac{1}{4} a^{3} + \frac{3}{2} a - \frac{1}{4}\) , \( a\) , \( -a^{3} + 2 a^{2} + 7 a - 16\) , \( -\frac{9}{2} a^{3} + 10 a^{2} + 33 a - \frac{145}{2}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{3}{2}a-\frac{1}{4}\right){x}^{2}+\left(-a^{3}+2a^{2}+7a-16\right){x}-\frac{9}{2}a^{3}+10a^{2}+33a-\frac{145}{2}$ |
| 25.1-b6 |
25.1-b |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1.469838141$ |
$5.175558822$ |
2.534507336 |
\( \frac{40501594119568875520256}{5} a^{3} - \frac{89106785247241445375744}{5} a^{2} - \frac{298080586628446171957273}{5} a + \frac{641231989324594775655826}{5} \) |
\( \bigl[1\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{23}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{19}{4}\) , \( 47 a^{3} + 52 a^{2} - 358 a - 425\) , \( 1387 a^{3} + 1499 a^{2} - 10752 a - 12668\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{19}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{23}{4}\right){x}^{2}+\left(47a^{3}+52a^{2}-358a-425\right){x}+1387a^{3}+1499a^{2}-10752a-12668$ |
| 25.1-c1 |
25.1-c |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.377877586$ |
$290.8077153$ |
2.288252076 |
\( \frac{196042139814484028621031}{20} a^{3} + \frac{89106785247241445375744}{5} a^{2} - \frac{477984375621386248191619}{10} a - \frac{264346966048807193473885}{4} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( -\frac{1}{4} a^{3} + \frac{1}{2} a - \frac{5}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( 300 a^{3} - 661 a^{2} - 2203 a + 4733\) , \( 16056 a^{3} - 35321 a^{2} - 118221 a + 254320\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{2}a-\frac{5}{4}\right){x}^{2}+\left(300a^{3}-661a^{2}-2203a+4733\right){x}+16056a^{3}-35321a^{2}-118221a+254320$ |
| 25.1-c2 |
25.1-c |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$0.094469396$ |
$290.8077153$ |
2.288252076 |
\( -\frac{10463690358987}{2500} a^{3} + \frac{9678146215168}{625} a^{2} + \frac{12827568843}{250} a - \frac{14724711948631}{500} \) |
\( \bigl[\frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( -\frac{1}{4} a^{3} - a^{2} + \frac{5}{2} a + \frac{15}{4}\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( -\frac{687}{4} a^{3} - 292 a^{2} + \frac{1835}{2} a + \frac{4905}{4}\) , \( \frac{9607}{4} a^{3} + 4492 a^{2} - \frac{22469}{2} a - \frac{63205}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}-a^{2}+\frac{5}{2}a+\frac{15}{4}\right){x}^{2}+\left(-\frac{687}{4}a^{3}-292a^{2}+\frac{1835}{2}a+\frac{4905}{4}\right){x}+\frac{9607}{4}a^{3}+4492a^{2}-\frac{22469}{2}a-\frac{63205}{4}$ |
| 25.1-c3 |
25.1-c |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$0.377877586$ |
$1163.230861$ |
2.288252076 |
\( \frac{716469}{20} a^{3} + \frac{328006}{5} a^{2} - \frac{1663381}{10} a - \frac{882663}{4} \) |
\( \bigl[a^{2} - 4\) , \( a^{2} - a - 4\) , \( a^{2} - 4\) , \( \frac{9}{4} a^{3} - 12 a^{2} + \frac{9}{2} a + \frac{93}{4}\) , \( \frac{31}{4} a^{3} - 30 a^{2} + \frac{3}{2} a + \frac{227}{4}\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(\frac{9}{4}a^{3}-12a^{2}+\frac{9}{2}a+\frac{93}{4}\right){x}+\frac{31}{4}a^{3}-30a^{2}+\frac{3}{2}a+\frac{227}{4}$ |
| 25.1-c4 |
25.1-c |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$0.188938793$ |
$1163.230861$ |
2.288252076 |
\( \frac{489915088671}{20} a^{3} + \frac{1113403817359}{25} a^{2} - \frac{5972487761641}{50} a - \frac{16515285091433}{100} \) |
\( \bigl[a^{2} - 4\) , \( a^{2} - a - 4\) , \( a^{2} - 4\) , \( 31 a^{3} - 167 a^{2} + 47 a + 347\) , \( -\frac{1957}{4} a^{3} + 1971 a^{2} - \frac{269}{2} a - \frac{15389}{4}\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(31a^{3}-167a^{2}+47a+347\right){x}-\frac{1957}{4}a^{3}+1971a^{2}-\frac{269}{2}a-\frac{15389}{4}$ |
| 25.1-c5 |
25.1-c |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.755755172$ |
$290.8077153$ |
2.288252076 |
\( \frac{23380795}{4} a^{3} - \frac{64261067}{5} a^{2} - \frac{430571273}{10} a + \frac{1851772771}{20} \) |
\( \bigl[a^{2} - 4\) , \( -\frac{1}{4} a^{3} + \frac{3}{2} a + \frac{3}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{15}{4}\) , \( -a^{3} + 6 a - 2\) , \( -\frac{1}{4} a^{3} + a^{2} + \frac{1}{2} a - \frac{41}{4}\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{15}{4}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{3}{2}a+\frac{3}{4}\right){x}^{2}+\left(-a^{3}+6a-2\right){x}-\frac{1}{4}a^{3}+a^{2}+\frac{1}{2}a-\frac{41}{4}$ |
| 25.1-c6 |
25.1-c |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.755755172$ |
$290.8077153$ |
2.288252076 |
\( -\frac{540595195}{4} a^{3} - \frac{730667933}{5} a^{2} + \frac{10473423273}{10} a + \frac{24675440989}{20} \) |
\( \bigl[1\) , \( \frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{19}{4}\) , \( \frac{1}{4} a^{3} - a^{2} - \frac{5}{2} a + \frac{25}{4}\) , \( -\frac{3}{2} a^{3} + 9 a - \frac{11}{2}\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{19}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){x}^{2}+\left(\frac{1}{4}a^{3}-a^{2}-\frac{5}{2}a+\frac{25}{4}\right){x}-\frac{3}{2}a^{3}+9a-\frac{11}{2}$ |
| 25.1-d1 |
25.1-d |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$0.734919070$ |
$1324.943058$ |
2.534507336 |
\( \frac{23380795}{4} a^{3} - \frac{64261067}{5} a^{2} - \frac{430571273}{10} a + \frac{1851772771}{20} \) |
\( \bigl[a\) , \( -\frac{1}{4} a^{3} - a^{2} + \frac{3}{2} a + \frac{23}{4}\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}\) , \( -\frac{3}{2} a^{3} - 4 a^{2} + 8 a + \frac{33}{2}\) , \( -\frac{23}{4} a^{3} - 11 a^{2} + \frac{57}{2} a + \frac{165}{4}\bigr] \) |
${y}^2+a{x}{y}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}-a^{2}+\frac{3}{2}a+\frac{23}{4}\right){x}^{2}+\left(-\frac{3}{2}a^{3}-4a^{2}+8a+\frac{33}{2}\right){x}-\frac{23}{4}a^{3}-11a^{2}+\frac{57}{2}a+\frac{165}{4}$ |
| 25.1-d2 |
25.1-d |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{8} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{2} \) |
$0.734919070$ |
$82.80894116$ |
2.534507336 |
\( \frac{489915088671}{20} a^{3} + \frac{1113403817359}{25} a^{2} - \frac{5972487761641}{50} a - \frac{16515285091433}{100} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{3}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( -\frac{3}{4} a^{3} - 3 a^{2} + \frac{53}{2} a - \frac{143}{4}\) , \( -2 a^{3} - 11 a^{2} + 77 a - 95\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{3}{4}\right){x}^{2}+\left(-\frac{3}{4}a^{3}-3a^{2}+\frac{53}{2}a-\frac{143}{4}\right){x}-2a^{3}-11a^{2}+77a-95$ |
| 25.1-d3 |
25.1-d |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2 \) |
$1.469838141$ |
$1324.943058$ |
2.534507336 |
\( \frac{716469}{20} a^{3} + \frac{328006}{5} a^{2} - \frac{1663381}{10} a - \frac{882663}{4} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{3}{2} a + \frac{1}{4}\) , \( \frac{1}{4} a^{3} - \frac{1}{2} a - \frac{3}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{19}{4}\) , \( -\frac{3}{4} a^{3} + 2 a^{2} + \frac{13}{2} a - \frac{63}{4}\) , \( \frac{5}{4} a^{3} - 3 a^{2} - \frac{23}{2} a + \frac{101}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{3}{2}a+\frac{1}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{19}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{3}{4}\right){x}^{2}+\left(-\frac{3}{4}a^{3}+2a^{2}+\frac{13}{2}a-\frac{63}{4}\right){x}+\frac{5}{4}a^{3}-3a^{2}-\frac{23}{2}a+\frac{101}{4}$ |
| 25.1-d4 |
25.1-d |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{2} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 1 \) |
$2.939676283$ |
$1324.943058$ |
2.534507336 |
\( -\frac{540595195}{4} a^{3} - \frac{730667933}{5} a^{2} + \frac{10473423273}{10} a + \frac{24675440989}{20} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}\) , \( -\frac{1}{4} a^{3} + a^{2} + \frac{1}{2} a - \frac{17}{4}\) , \( a^{2} - 4\) , \( -\frac{5}{2} a^{3} + 7 a + \frac{15}{2}\) , \( -\frac{17}{4} a^{3} - 2 a^{2} + \frac{33}{2} a + \frac{51}{4}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+a^{2}+\frac{1}{2}a-\frac{17}{4}\right){x}^{2}+\left(-\frac{5}{2}a^{3}+7a+\frac{15}{2}\right){x}-\frac{17}{4}a^{3}-2a^{2}+\frac{33}{2}a+\frac{51}{4}$ |
| 25.1-d5 |
25.1-d |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{16} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1.469838141$ |
$5.175558822$ |
2.534507336 |
\( -\frac{10463690358987}{2500} a^{3} + \frac{9678146215168}{625} a^{2} + \frac{12827568843}{250} a - \frac{14724711948631}{500} \) |
\( \bigl[\frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{15}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{3}{2} a - \frac{15}{4}\) , \( -\frac{241}{4} a^{3} - 53 a^{2} + \frac{915}{2} a + \frac{1799}{4}\) , \( \frac{707}{2} a^{3} + 404 a^{2} - 2799 a - \frac{7087}{2}\bigr] \) |
${y}^2+\left(\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}\right){x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{15}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{3}{2}a-\frac{15}{4}\right){x}^{2}+\left(-\frac{241}{4}a^{3}-53a^{2}+\frac{915}{2}a+\frac{1799}{4}\right){x}+\frac{707}{2}a^{3}+404a^{2}-2799a-\frac{7087}{2}$ |
| 25.1-d6 |
25.1-d |
$6$ |
$8$ |
4.4.9225.1 |
$4$ |
$[4, 0]$ |
25.1 |
\( 5^{2} \) |
\( 5^{4} \) |
$12.83407$ |
$(1/2a^3-3a-1/2)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1.469838141$ |
$5.175558822$ |
2.534507336 |
\( \frac{196042139814484028621031}{20} a^{3} + \frac{89106785247241445375744}{5} a^{2} - \frac{477984375621386248191619}{10} a - \frac{264346966048807193473885}{4} \) |
\( \bigl[1\) , \( \frac{1}{4} a^{3} - a^{2} - \frac{5}{2} a + \frac{21}{4}\) , \( \frac{1}{4} a^{3} + a^{2} - \frac{1}{2} a - \frac{15}{4}\) , \( 15 a^{3} - 53 a^{2} - 15 a + 120\) , \( \frac{809}{2} a^{3} - 1502 a^{2} + a + \frac{5709}{2}\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{4}a^{3}+a^{2}-\frac{1}{2}a-\frac{15}{4}\right){y}={x}^{3}+\left(\frac{1}{4}a^{3}-a^{2}-\frac{5}{2}a+\frac{21}{4}\right){x}^{2}+\left(15a^{3}-53a^{2}-15a+120\right){x}+\frac{809}{2}a^{3}-1502a^{2}+a+\frac{5709}{2}$ |
*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.
