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
5.1-a1
5.1-a
$1$
$1$
5.5.65657.1
$5$
$[5, 0]$
5.1
\( 5 \)
\( -5 \)
$26.89528$
$(-a^2+a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 1 \)
$1$
$526.9651791$
2.05656008
\( \frac{144184611}{5} a^{4} - \frac{255843129}{5} a^{3} - \frac{524006273}{5} a^{2} + \frac{693618211}{5} a + \frac{186593617}{5} \)
\( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 3\) , \( -a^{4} + a^{3} + 4 a^{2} - 3 a - 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( -3 a^{4} + 2 a^{3} + 10 a^{2} - 7 a - 3\) , \( a^{4} + 3 a^{3} - 6 a - 3\bigr] \)
${y}^2+\left(-a^{4}+2a^{3}+4a^{2}-5a-3\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){y}={x}^{3}+\left(-a^{4}+a^{3}+4a^{2}-3a-1\right){x}^{2}+\left(-3a^{4}+2a^{3}+10a^{2}-7a-3\right){x}+a^{4}+3a^{3}-6a-3$
25.1-b1
25.1-b
$1$
$1$
5.5.65657.1
$5$
$[5, 0]$
25.1
\( 5^{2} \)
\( - 5^{7} \)
$31.59171$
$(-a^2+a+2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$1$
\( 2^{2} \)
$0.002247456$
$8560.393133$
1.50167015
\( \frac{144184611}{5} a^{4} - \frac{255843129}{5} a^{3} - \frac{524006273}{5} a^{2} + \frac{693618211}{5} a + \frac{186593617}{5} \)
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 3 a + 3\) , \( -a^{4} + 5 a^{2} - 2\) , \( -a^{4} + a^{3} + 5 a^{2} - 2 a - 3\) , \( 18 a^{4} - 34 a^{3} - 65 a^{2} + 92 a + 27\) , \( -50 a^{4} + 87 a^{3} + 183 a^{2} - 235 a - 65\bigr] \)
${y}^2+\left(a^{4}-a^{3}-4a^{2}+3a+3\right){x}{y}+\left(-a^{4}+a^{3}+5a^{2}-2a-3\right){y}={x}^{3}+\left(-a^{4}+5a^{2}-2\right){x}^{2}+\left(18a^{4}-34a^{3}-65a^{2}+92a+27\right){x}-50a^{4}+87a^{3}+183a^{2}-235a-65$
45.1-d1
45.1-d
$1$
$1$
5.5.65657.1
$5$
$[5, 0]$
45.1
\( 3^{2} \cdot 5 \)
\( - 3^{6} \cdot 5 \)
$33.50428$
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$1$
\( 1 \)
$1$
$325.0490427$
1.26855229
\( \frac{144184611}{5} a^{4} - \frac{255843129}{5} a^{3} - \frac{524006273}{5} a^{2} + \frac{693618211}{5} a + \frac{186593617}{5} \)
\( \bigl[-2 a^{4} + 3 a^{3} + 9 a^{2} - 8 a - 7\) , \( -3 a^{4} + 4 a^{3} + 13 a^{2} - 12 a - 9\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 3\) , \( -7 a^{4} + 2 a^{3} + 20 a^{2} - 11 a - 2\) , \( -11 a^{4} - 8 a^{3} + 24 a^{2} + 9 a + 2\bigr] \)
${y}^2+\left(-2a^{4}+3a^{3}+9a^{2}-8a-7\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+3\right){y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-12a-9\right){x}^{2}+\left(-7a^{4}+2a^{3}+20a^{2}-11a-2\right){x}-11a^{4}-8a^{3}+24a^{2}+9a+2$
225.1-g1
225.1-g
$1$
$1$
5.5.65657.1
$5$
$[5, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( - 3^{6} \cdot 5^{7} \)
$39.35476$
$(-a^4+a^3+4a^2-2a-2), (-a^2+a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$1$
\( 2 \)
$1$
$249.2635278$
1.94557607
\( \frac{144184611}{5} a^{4} - \frac{255843129}{5} a^{3} - \frac{524006273}{5} a^{2} + \frac{693618211}{5} a + \frac{186593617}{5} \)
\( \bigl[a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( a^{4} - a^{3} - 5 a^{2} + 2 a + 5\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 6 a - 5\) , \( -2 a^{3} + 4 a - 1\) , \( -17 a^{4} + 19 a^{3} + 77 a^{2} - 49 a - 68\bigr] \)
${y}^2+\left(a^{4}-a^{3}-4a^{2}+2a+3\right){x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-6a-5\right){y}={x}^{3}+\left(a^{4}-a^{3}-5a^{2}+2a+5\right){x}^{2}+\left(-2a^{3}+4a-1\right){x}-17a^{4}+19a^{3}+77a^{2}-49a-68$
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*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.