| 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 |
| 45.1-a1 |
45.1-a |
$1$ |
$1$ |
4.4.13525.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5 \) |
$16.72472$ |
$(3/5a^3+a^2-16/5a-22/5), (-2/5a^3-a^2+19/5a+43/5)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$463.6560988$ |
3.986826533 |
\( -\frac{114023}{5} a^{3} + \frac{405149}{15} a^{2} + \frac{5219281}{15} a - \frac{9956996}{15} \) |
\( \bigl[a + 1\) , \( a^{2} + a - 6\) , \( a^{2} - 6\) , \( \frac{9}{5} a^{3} - \frac{33}{5} a + \frac{4}{5}\) , \( \frac{4}{5} a^{3} + 9 a^{2} - \frac{43}{5} a - \frac{171}{5}\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-6\right){y}={x}^{3}+\left(a^{2}+a-6\right){x}^{2}+\left(\frac{9}{5}a^{3}-\frac{33}{5}a+\frac{4}{5}\right){x}+\frac{4}{5}a^{3}+9a^{2}-\frac{43}{5}a-\frac{171}{5}$ |
| 45.1-b1 |
45.1-b |
$2$ |
$2$ |
4.4.13525.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( - 3^{4} \cdot 5^{4} \) |
$16.72472$ |
$(3/5a^3+a^2-16/5a-22/5), (-2/5a^3-a^2+19/5a+43/5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$46.97125407$ |
3.231123110 |
\( \frac{3826912461601}{225} a^{3} - \frac{376124214242}{225} a^{2} - \frac{3002241843276}{25} a + \frac{725754518578}{9} \) |
\( \bigl[\frac{1}{5} a^{3} - \frac{7}{5} a + \frac{1}{5}\) , \( -\frac{1}{5} a^{3} + \frac{2}{5} a - \frac{6}{5}\) , \( a^{2} + a - 5\) , \( -\frac{507}{5} a^{3} + 249 a^{2} + \frac{4234}{5} a - \frac{10107}{5}\) , \( \frac{9727}{5} a^{3} - 4806 a^{2} - \frac{80984}{5} a + \frac{194342}{5}\bigr] \) |
${y}^2+\left(\frac{1}{5}a^{3}-\frac{7}{5}a+\frac{1}{5}\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-\frac{1}{5}a^{3}+\frac{2}{5}a-\frac{6}{5}\right){x}^{2}+\left(-\frac{507}{5}a^{3}+249a^{2}+\frac{4234}{5}a-\frac{10107}{5}\right){x}+\frac{9727}{5}a^{3}-4806a^{2}-\frac{80984}{5}a+\frac{194342}{5}$ |
| 45.1-b2 |
45.1-b |
$2$ |
$2$ |
4.4.13525.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{2} \cdot 5^{8} \) |
$16.72472$ |
$(3/5a^3+a^2-16/5a-22/5), (-2/5a^3-a^2+19/5a+43/5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$187.8850162$ |
3.231123110 |
\( \frac{4408151}{1875} a^{3} + \frac{166648333}{1875} a^{2} + \frac{64439341}{1875} a - \frac{14311189}{25} \) |
\( \bigl[\frac{1}{5} a^{3} - \frac{7}{5} a + \frac{1}{5}\) , \( -\frac{1}{5} a^{3} + \frac{2}{5} a - \frac{6}{5}\) , \( a^{2} + a - 5\) , \( -\frac{17}{5} a^{3} + 4 a^{2} + \frac{254}{5} a - \frac{467}{5}\) , \( \frac{218}{5} a^{3} - 130 a^{2} - \frac{1321}{5} a + \frac{3798}{5}\bigr] \) |
${y}^2+\left(\frac{1}{5}a^{3}-\frac{7}{5}a+\frac{1}{5}\right){x}{y}+\left(a^{2}+a-5\right){y}={x}^{3}+\left(-\frac{1}{5}a^{3}+\frac{2}{5}a-\frac{6}{5}\right){x}^{2}+\left(-\frac{17}{5}a^{3}+4a^{2}+\frac{254}{5}a-\frac{467}{5}\right){x}+\frac{218}{5}a^{3}-130a^{2}-\frac{1321}{5}a+\frac{3798}{5}$ |
| 45.1-c1 |
45.1-c |
$2$ |
$2$ |
4.4.13525.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{6} \cdot 5^{2} \) |
$16.72472$ |
$(3/5a^3+a^2-16/5a-22/5), (-2/5a^3-a^2+19/5a+43/5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$146.8909155$ |
1.894600117 |
\( -\frac{873720511}{45} a^{3} + \frac{10471578184}{135} a^{2} + \frac{1596010592}{135} a - \frac{27027109603}{135} \) |
\( \bigl[\frac{1}{5} a^{3} + a^{2} - \frac{7}{5} a - \frac{29}{5}\) , \( -\frac{1}{5} a^{3} + a^{2} + \frac{12}{5} a - \frac{36}{5}\) , \( \frac{1}{5} a^{3} + a^{2} - \frac{2}{5} a - \frac{24}{5}\) , \( \frac{6}{5} a^{3} - 3 a^{2} - \frac{27}{5} a + \frac{76}{5}\) , \( \frac{3}{5} a^{3} - 2 a^{2} + \frac{14}{5} a - \frac{22}{5}\bigr] \) |
${y}^2+\left(\frac{1}{5}a^{3}+a^{2}-\frac{7}{5}a-\frac{29}{5}\right){x}{y}+\left(\frac{1}{5}a^{3}+a^{2}-\frac{2}{5}a-\frac{24}{5}\right){y}={x}^{3}+\left(-\frac{1}{5}a^{3}+a^{2}+\frac{12}{5}a-\frac{36}{5}\right){x}^{2}+\left(\frac{6}{5}a^{3}-3a^{2}-\frac{27}{5}a+\frac{76}{5}\right){x}+\frac{3}{5}a^{3}-2a^{2}+\frac{14}{5}a-\frac{22}{5}$ |
| 45.1-c2 |
45.1-c |
$2$ |
$2$ |
4.4.13525.1 |
$4$ |
$[4, 0]$ |
45.1 |
\( 3^{2} \cdot 5 \) |
\( 3^{12} \cdot 5 \) |
$16.72472$ |
$(3/5a^3+a^2-16/5a-22/5), (-2/5a^3-a^2+19/5a+43/5)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$146.8909155$ |
1.894600117 |
\( \frac{7150298296431604}{3645} a^{3} + \frac{14681250308973859}{3645} a^{2} - \frac{910629148427686}{81} a - \frac{67914310580967908}{3645} \) |
\( \bigl[\frac{1}{5} a^{3} + a^{2} - \frac{7}{5} a - \frac{29}{5}\) , \( -\frac{1}{5} a^{3} + a^{2} + \frac{12}{5} a - \frac{36}{5}\) , \( \frac{1}{5} a^{3} + a^{2} - \frac{2}{5} a - \frac{24}{5}\) , \( -\frac{14}{5} a^{3} - 8 a^{2} + \frac{88}{5} a + \frac{131}{5}\) , \( \frac{32}{5} a^{3} - 4 a^{2} - \frac{214}{5} a + \frac{112}{5}\bigr] \) |
${y}^2+\left(\frac{1}{5}a^{3}+a^{2}-\frac{7}{5}a-\frac{29}{5}\right){x}{y}+\left(\frac{1}{5}a^{3}+a^{2}-\frac{2}{5}a-\frac{24}{5}\right){y}={x}^{3}+\left(-\frac{1}{5}a^{3}+a^{2}+\frac{12}{5}a-\frac{36}{5}\right){x}^{2}+\left(-\frac{14}{5}a^{3}-8a^{2}+\frac{88}{5}a+\frac{131}{5}\right){x}+\frac{32}{5}a^{3}-4a^{2}-\frac{214}{5}a+\frac{112}{5}$ |
*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.
