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 |
45.4-a1 |
45.4-a |
$2$ |
$2$ |
4.4.4525.1 |
$4$ |
$[4, 0]$ |
45.4 |
\( 3^{2} \cdot 5 \) |
\( - 3^{4} \cdot 5^{8} \) |
$9.67385$ |
$(-a^2+a+5), (1/3a^3-1/3a^2-7/3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{4} \) |
$0.451118733$ |
$24.99201932$ |
2.681654982 |
\( -\frac{61709015057278}{1875} a^{3} + \frac{568055547171271}{5625} a^{2} + \frac{40298462678968}{1875} a - \frac{17899615289469}{125} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 4\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 4\) , \( a + 1\) , \( 7 a^{3} + a^{2} - 53 a - 53\) , \( 68 a^{3} + 20 a^{2} - 466 a - 448\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-4\right){x}^{2}+\left(7a^{3}+a^{2}-53a-53\right){x}+68a^{3}+20a^{2}-466a-448$ |
45.4-a2 |
45.4-a |
$2$ |
$2$ |
4.4.4525.1 |
$4$ |
$[4, 0]$ |
45.4 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{4} \) |
$9.67385$ |
$(-a^2+a+5), (1/3a^3-1/3a^2-7/3a+1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.902237467$ |
$49.98403864$ |
2.681654982 |
\( -\frac{1262963}{25} a^{3} - \frac{3615727}{25} a^{2} + \frac{51598387}{75} a + \frac{21590004}{25} \) |
\( \bigl[1\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 4\) , \( 1\) , \( -\frac{65}{3} a^{3} + \frac{155}{3} a^{2} + \frac{296}{3} a - 209\) , \( -156 a^{3} + 334 a^{2} + 727 a - 1306\bigr] \) |
${y}^2+{x}{y}+{y}={x}^{3}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-4\right){x}^{2}+\left(-\frac{65}{3}a^{3}+\frac{155}{3}a^{2}+\frac{296}{3}a-209\right){x}-156a^{3}+334a^{2}+727a-1306$ |
45.4-b1 |
45.4-b |
$2$ |
$5$ |
4.4.4525.1 |
$4$ |
$[4, 0]$ |
45.4 |
\( 3^{2} \cdot 5 \) |
\( 3^{10} \cdot 5 \) |
$9.67385$ |
$(-a^2+a+5), (1/3a^3-1/3a^2-7/3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1[2] |
$1$ |
\( 5 \) |
$0.015271155$ |
$472.5674111$ |
2.145637649 |
\( \frac{311063}{405} a^{3} + \frac{426403}{243} a^{2} - \frac{2175776}{405} a - \frac{1637564}{135} \) |
\( \bigl[a^{2} - a - 4\) , \( -a^{2} + a + 5\) , \( \frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{4}{3} a - 3\) , \( -\frac{44}{3} a^{3} + \frac{86}{3} a^{2} + \frac{209}{3} a - 108\) , \( 56 a^{3} - 120 a^{2} - 262 a + 467\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{4}{3}a-3\right){y}={x}^{3}+\left(-a^{2}+a+5\right){x}^{2}+\left(-\frac{44}{3}a^{3}+\frac{86}{3}a^{2}+\frac{209}{3}a-108\right){x}+56a^{3}-120a^{2}-262a+467$ |
45.4-b2 |
45.4-b |
$2$ |
$5$ |
4.4.4525.1 |
$4$ |
$[4, 0]$ |
45.4 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{5} \) |
$9.67385$ |
$(-a^2+a+5), (1/3a^3-1/3a^2-7/3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1[2] |
$1$ |
\( 1 \) |
$0.076355779$ |
$472.5674111$ |
2.145637649 |
\( -\frac{23573637}{125} a^{3} - \frac{24357233}{75} a^{2} + \frac{54667564}{125} a + \frac{78032973}{125} \) |
\( \bigl[1\) , \( a^{2} - a - 4\) , \( \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{4}{3} a + 1\) , \( -a^{3} + 5 a + 4\) , \( \frac{20}{3} a^{3} + \frac{10}{3} a^{2} - \frac{128}{3} a - 43\bigr] \) |
${y}^2+{x}{y}+\left(\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{4}{3}a+1\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-a^{3}+5a+4\right){x}+\frac{20}{3}a^{3}+\frac{10}{3}a^{2}-\frac{128}{3}a-43$ |
45.4-c1 |
45.4-c |
$2$ |
$3$ |
4.4.4525.1 |
$4$ |
$[4, 0]$ |
45.4 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{3} \) |
$9.67385$ |
$(-a^2+a+5), (1/3a^3-1/3a^2-7/3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 3 \) |
$0.007009552$ |
$1579.532167$ |
1.975107543 |
\( \frac{95132237}{225} a^{3} - \frac{217836256}{135} a^{2} + \frac{232697396}{225} a + \frac{45330989}{75} \) |
\( \bigl[\frac{1}{3} a^{3} + \frac{2}{3} a^{2} - \frac{7}{3} a - 4\) , \( -\frac{1}{3} a^{3} + \frac{4}{3} a^{2} + \frac{4}{3} a - 3\) , \( a^{2} - 4\) , \( -\frac{4}{3} a^{3} + \frac{1}{3} a^{2} + \frac{25}{3} a + 4\) , \( -\frac{5}{3} a^{3} - \frac{4}{3} a^{2} + \frac{32}{3} a + 13\bigr] \) |
${y}^2+\left(\frac{1}{3}a^{3}+\frac{2}{3}a^{2}-\frac{7}{3}a-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-\frac{1}{3}a^{3}+\frac{4}{3}a^{2}+\frac{4}{3}a-3\right){x}^{2}+\left(-\frac{4}{3}a^{3}+\frac{1}{3}a^{2}+\frac{25}{3}a+4\right){x}-\frac{5}{3}a^{3}-\frac{4}{3}a^{2}+\frac{32}{3}a+13$ |
45.4-c2 |
45.4-c |
$2$ |
$3$ |
4.4.4525.1 |
$4$ |
$[4, 0]$ |
45.4 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5 \) |
$9.67385$ |
$(-a^2+a+5), (1/3a^3-1/3a^2-7/3a+1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.021028658$ |
$1579.532167$ |
1.975107543 |
\( -\frac{1037039198606}{15} a^{3} - 118962068606 a^{2} + \frac{2404382003012}{15} a + \frac{1143498460323}{5} \) |
\( \bigl[a^{2} - 3\) , \( \frac{1}{3} a^{3} - \frac{4}{3} a^{2} - \frac{1}{3} a + 3\) , \( a\) , \( \frac{10}{3} a^{3} + \frac{8}{3} a^{2} - \frac{73}{3} a - 24\) , \( -8 a^{3} - 6 a^{2} + 53 a + 60\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(\frac{1}{3}a^{3}-\frac{4}{3}a^{2}-\frac{1}{3}a+3\right){x}^{2}+\left(\frac{10}{3}a^{3}+\frac{8}{3}a^{2}-\frac{73}{3}a-24\right){x}-8a^{3}-6a^{2}+53a+60$ |
*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.