| 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 |
| 36.1-a1 |
36.1-a |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{10} \cdot 3^{2} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$128.7182515$ |
1.300781732 |
\( -\frac{2240392722766771819}{8} a^{3} - \frac{5522297380891621979}{12} a^{2} + \frac{3630089700724133759}{12} a + \frac{13074436924466614051}{24} \) |
\( \bigl[-a^{3} + 4 a^{2} + 2 a - 8\) , \( a^{3} - 4 a^{2} + 7\) , \( a^{3} - 3 a^{2} - 2 a + 4\) , \( -545 a^{3} + 1664 a^{2} + 2046 a - 3391\) , \( 11445 a^{3} - 35734 a^{2} - 40291 a + 69868\bigr] \) |
${y}^2+\left(-a^{3}+4a^{2}+2a-8\right){x}{y}+\left(a^{3}-3a^{2}-2a+4\right){y}={x}^{3}+\left(a^{3}-4a^{2}+7\right){x}^{2}+\left(-545a^{3}+1664a^{2}+2046a-3391\right){x}+11445a^{3}-35734a^{2}-40291a+69868$ |
| 36.1-a2 |
36.1-a |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{2} \cdot 3^{10} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$128.7182515$ |
1.300781732 |
\( \frac{7769042107}{27} a^{3} - \frac{4821837803}{18} a^{2} - \frac{124228493479}{54} a - \frac{101732420597}{54} \) |
\( \bigl[-a^{3} + 4 a^{2} + 2 a - 8\) , \( a^{3} - 4 a^{2} + 7\) , \( a^{3} - 3 a^{2} - 2 a + 4\) , \( -10 a^{3} + 34 a^{2} + 26 a - 61\) , \( -15 a^{3} + 55 a^{2} + 33 a - 124\bigr] \) |
${y}^2+\left(-a^{3}+4a^{2}+2a-8\right){x}{y}+\left(a^{3}-3a^{2}-2a+4\right){y}={x}^{3}+\left(a^{3}-4a^{2}+7\right){x}^{2}+\left(-10a^{3}+34a^{2}+26a-61\right){x}-15a^{3}+55a^{2}+33a-124$ |
| 36.1-b1 |
36.1-b |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{20} \cdot 3^{10} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$3.407084799$ |
1.721540502 |
\( \frac{76847592208429853}{864} a^{3} + \frac{125275986986772941}{864} a^{2} - \frac{577484854103149}{6} a - \frac{148183293619169353}{864} \) |
\( \bigl[a^{2} - 2 a - 4\) , \( -a^{3} + 3 a^{2} + 2 a - 4\) , \( a + 1\) , \( -108 a^{3} + 370 a^{2} + 226 a - 529\) , \( 1447 a^{3} - 5049 a^{2} - 2590 a + 6708\bigr] \) |
${y}^2+\left(a^{2}-2a-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-4\right){x}^{2}+\left(-108a^{3}+370a^{2}+226a-529\right){x}+1447a^{3}-5049a^{2}-2590a+6708$ |
| 36.1-b2 |
36.1-b |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{4} \cdot 3^{2} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$2129.427999$ |
1.721540502 |
\( \frac{98225}{6} a^{3} - \frac{170425}{3} a^{2} - \frac{180341}{6} a + \frac{153763}{2} \) |
\( \bigl[a^{2} - 2 a - 3\) , \( -2 a^{3} + 7 a^{2} + 3 a - 13\) , \( a\) , \( -3 a^{3} + 9 a^{2} + 5 a - 10\) , \( -2 a^{3} + 7 a^{2} + 4 a - 10\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){x}{y}+a{y}={x}^{3}+\left(-2a^{3}+7a^{2}+3a-13\right){x}^{2}+\left(-3a^{3}+9a^{2}+5a-10\right){x}-2a^{3}+7a^{2}+4a-10$ |
| 36.1-c1 |
36.1-c |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{20} \cdot 3^{10} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5 \) |
$0.040106866$ |
$192.6722033$ |
3.123648015 |
\( \frac{76847592208429853}{864} a^{3} + \frac{125275986986772941}{864} a^{2} - \frac{577484854103149}{6} a - \frac{148183293619169353}{864} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 5\) , \( a^{2} - a - 3\) , \( a\) , \( 22 a^{3} - 5 a^{2} - 206 a - 223\) , \( -357 a^{3} + 303 a^{2} + 2919 a + 2498\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+5\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(22a^{3}-5a^{2}-206a-223\right){x}-357a^{3}+303a^{2}+2919a+2498$ |
| 36.1-c2 |
36.1-c |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{4} \cdot 3^{2} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$0.200534334$ |
$192.6722033$ |
3.123648015 |
\( \frac{98225}{6} a^{3} - \frac{170425}{3} a^{2} - \frac{180341}{6} a + \frac{153763}{2} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 5\) , \( a^{2} - a - 3\) , \( a\) , \( 2 a^{3} - 6 a - 3\) , \( 4 a^{3} + 2 a^{2} - 17 a - 17\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+5\right){x}{y}+a{y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(2a^{3}-6a-3\right){x}+4a^{3}+2a^{2}-17a-17$ |
| 36.1-d1 |
36.1-d |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{10} \cdot 3^{2} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$8.717478747$ |
$0.362426267$ |
3.192823130 |
\( -\frac{2240392722766771819}{8} a^{3} - \frac{5522297380891621979}{12} a^{2} + \frac{3630089700724133759}{12} a + \frac{13074436924466614051}{24} \) |
\( \bigl[1\) , \( a^{3} - 3 a^{2} - 2 a + 5\) , \( 0\) , \( -1420 a^{3} + 1299 a^{2} + 11345 a + 9299\) , \( -199304 a^{3} + 185398 a^{2} + 1593168 a + 1304734\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(a^{3}-3a^{2}-2a+5\right){x}^{2}+\left(-1420a^{3}+1299a^{2}+11345a+9299\right){x}-199304a^{3}+185398a^{2}+1593168a+1304734$ |
| 36.1-d2 |
36.1-d |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{2} \cdot 3^{10} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1.743495749$ |
$226.5164170$ |
3.192823130 |
\( \frac{7769042107}{27} a^{3} - \frac{4821837803}{18} a^{2} - \frac{124228493479}{54} a - \frac{101732420597}{54} \) |
\( \bigl[a + 1\) , \( a^{2} - 2 a - 4\) , \( 0\) , \( -2 a^{3} + 5 a^{2} + 8 a - 7\) , \( a^{3} - 11 a^{2} - 6 a + 16\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(-2a^{3}+5a^{2}+8a-7\right){x}+a^{3}-11a^{2}-6a+16$ |
| 36.1-e1 |
36.1-e |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{10} \cdot 3^{2} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1$ |
$128.7182515$ |
1.300781732 |
\( -\frac{24076175056414959271}{4} a^{3} + \frac{464579280282153457207}{24} a^{2} + \frac{446274506118921945731}{24} a - \frac{831645570259901538439}{24} \) |
\( \bigl[a^{2} - a - 4\) , \( -2 a^{3} + 7 a^{2} + 4 a - 11\) , \( 1\) , \( -18 a^{3} + 27 a^{2} - 358 a - 617\) , \( 71 a^{3} + 1113 a^{2} + 4748 a + 4759\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+{y}={x}^{3}+\left(-2a^{3}+7a^{2}+4a-11\right){x}^{2}+\left(-18a^{3}+27a^{2}-358a-617\right){x}+71a^{3}+1113a^{2}+4748a+4759$ |
| 36.1-e2 |
36.1-e |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{2} \cdot 3^{10} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$1$ |
$128.7182515$ |
1.300781732 |
\( \frac{66518800309}{54} a^{3} - \frac{38617523360}{9} a^{2} - \frac{60971080045}{27} a + \frac{313914856043}{54} \) |
\( \bigl[a^{2} - a - 4\) , \( -2 a^{3} + 7 a^{2} + 4 a - 11\) , \( 1\) , \( 2 a^{3} - 8 a^{2} - 3 a + 13\) , \( 11 a^{3} - 46 a^{2} - 21 a + 62\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+{y}={x}^{3}+\left(-2a^{3}+7a^{2}+4a-11\right){x}^{2}+\left(2a^{3}-8a^{2}-3a+13\right){x}+11a^{3}-46a^{2}-21a+62$ |
| 36.1-f1 |
36.1-f |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{4} \cdot 3^{2} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$2129.427999$ |
1.721540502 |
\( \frac{27859}{6} a^{3} - \frac{18701}{3} a^{2} - \frac{197911}{6} a - 20236 \) |
\( \bigl[a^{2} - 2 a - 4\) , \( -a^{3} + 3 a^{2} + 3 a - 3\) , \( -a^{3} + 4 a^{2} + 2 a - 7\) , \( -7 a^{3} + 19 a^{2} + 28 a - 22\) , \( -5 a^{3} + 12 a^{2} + 23 a - 9\bigr] \) |
${y}^2+\left(a^{2}-2a-4\right){x}{y}+\left(-a^{3}+4a^{2}+2a-7\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+3a-3\right){x}^{2}+\left(-7a^{3}+19a^{2}+28a-22\right){x}-5a^{3}+12a^{2}+23a-9$ |
| 36.1-f2 |
36.1-f |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{20} \cdot 3^{10} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 2 \) |
$1$ |
$3.407084799$ |
1.721540502 |
\( \frac{411876901072778989}{216} a^{3} - \frac{165573174265169074}{27} a^{2} - \frac{565545307834198219}{96} a + \frac{4742105292427286831}{432} \) |
\( \bigl[a^{2} - 2 a - 3\) , \( a^{2} - 2 a - 4\) , \( a^{2} - 2 a - 3\) , \( -16 a^{3} + a^{2} + 149 a + 150\) , \( 392 a^{3} - 414 a^{2} - 3077 a - 2425\bigr] \) |
${y}^2+\left(a^{2}-2a-3\right){x}{y}+\left(a^{2}-2a-3\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(-16a^{3}+a^{2}+149a+150\right){x}+392a^{3}-414a^{2}-3077a-2425$ |
| 36.1-g1 |
36.1-g |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{4} \cdot 3^{2} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 2 \) |
$0.200534334$ |
$192.6722033$ |
3.123648015 |
\( \frac{27859}{6} a^{3} - \frac{18701}{3} a^{2} - \frac{197911}{6} a - 20236 \) |
\( \bigl[a\) , \( a^{2} - 2 a - 4\) , \( a + 1\) , \( 2 a^{3} - 8 a^{2} - a + 14\) , \( 12 a^{3} - 41 a^{2} - 24 a + 53\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(2a^{3}-8a^{2}-a+14\right){x}+12a^{3}-41a^{2}-24a+53$ |
| 36.1-g2 |
36.1-g |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{20} \cdot 3^{10} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 2 \cdot 5 \) |
$0.040106866$ |
$192.6722033$ |
3.123648015 |
\( \frac{411876901072778989}{216} a^{3} - \frac{165573174265169074}{27} a^{2} - \frac{565545307834198219}{96} a + \frac{4742105292427286831}{432} \) |
\( \bigl[a\) , \( a^{2} - 2 a - 4\) , \( a + 1\) , \( 102 a^{3} - 363 a^{2} - 161 a + 469\) , \( -1349 a^{3} + 4714 a^{2} + 2406 a - 6317\bigr] \) |
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a^{2}-2a-4\right){x}^{2}+\left(102a^{3}-363a^{2}-161a+469\right){x}-1349a^{3}+4714a^{2}+2406a-6317$ |
| 36.1-h1 |
36.1-h |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{10} \cdot 3^{2} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$8.717478747$ |
$0.362426267$ |
3.192823130 |
\( -\frac{24076175056414959271}{4} a^{3} + \frac{464579280282153457207}{24} a^{2} + \frac{446274506118921945731}{24} a - \frac{831645570259901538439}{24} \) |
\( \bigl[1\) , \( -a^{3} + 3 a^{2} + 2 a - 5\) , \( a^{3} - 3 a^{2} - 3 a + 5\) , \( -6184 a^{3} + 21515 a^{2} + 11467 a - 29331\) , \( -856340 a^{3} + 2982695 a^{2} + 1570802 a - 4042118\bigr] \) |
${y}^2+{x}{y}+\left(a^{3}-3a^{2}-3a+5\right){y}={x}^{3}+\left(-a^{3}+3a^{2}+2a-5\right){x}^{2}+\left(-6184a^{3}+21515a^{2}+11467a-29331\right){x}-856340a^{3}+2982695a^{2}+1570802a-4042118$ |
| 36.1-h2 |
36.1-h |
$2$ |
$5$ |
4.4.9792.1 |
$4$ |
$[4, 0]$ |
36.1 |
\( 2^{2} \cdot 3^{2} \) |
\( - 2^{2} \cdot 3^{10} \) |
$13.83923$ |
$(a^3-3a^2-3a+4), (a^3-3a^2-2a+5)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1.743495749$ |
$226.5164170$ |
3.192823130 |
\( \frac{66518800309}{54} a^{3} - \frac{38617523360}{9} a^{2} - \frac{60971080045}{27} a + \frac{313914856043}{54} \) |
\( \bigl[a^{3} - 3 a^{2} - 2 a + 4\) , \( -2 a^{3} + 7 a^{2} + 5 a - 13\) , \( a\) , \( -6 a^{3} + 20 a^{2} + 18 a - 35\) , \( -6 a^{3} + 20 a^{2} + 18 a - 36\bigr] \) |
${y}^2+\left(a^{3}-3a^{2}-2a+4\right){x}{y}+a{y}={x}^{3}+\left(-2a^{3}+7a^{2}+5a-13\right){x}^{2}+\left(-6a^{3}+20a^{2}+18a-35\right){x}-6a^{3}+20a^{2}+18a-36$ |
*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.
