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-b2
5.1-b
$4$
$15$
5.5.65657.1
$5$
$[5, 0]$
5.1
\( 5 \)
\( -5 \)
$26.89528$
$(-a^2+a+2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$3, 5$
3B , 5B.1.2
$625$
\( 1 \)
$1$
$0.333588330$
0.813673832
\( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \)
\( \bigl[-a^{4} + 2 a^{3} + 4 a^{2} - 5 a - 2\) , \( -2 a^{4} + 3 a^{3} + 9 a^{2} - 10 a - 8\) , \( a^{4} - a^{3} - 4 a^{2} + 2 a + 3\) , \( 86 a^{4} + 73 a^{3} - 426 a^{2} - 515 a - 104\) , \( 1231 a^{4} + 500 a^{3} - 5653 a^{2} - 5184 a - 905\bigr] \)
${y}^2+\left(-a^{4}+2a^{3}+4a^{2}-5a-2\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+2a+3\right){y}={x}^{3}+\left(-2a^{4}+3a^{3}+9a^{2}-10a-8\right){x}^{2}+\left(86a^{4}+73a^{3}-426a^{2}-515a-104\right){x}+1231a^{4}+500a^{3}-5653a^{2}-5184a-905$
25.1-c2
25.1-c
$4$
$15$
5.5.65657.1
$5$
$[5, 0]$
25.1
\( 5^{2} \)
\( - 5^{7} \)
$31.59171$
$(-a^2+a+2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3, 5$
3B , 5B.4.2
$1$
\( 2^{2} \)
$0.223493896$
$163.1635062$
2.84628363
\( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \)
\( \bigl[-a^{4} + a^{3} + 5 a^{2} - 3 a - 4\) , \( a - 1\) , \( a^{4} - a^{3} - 4 a^{2} + 3 a + 2\) , \( -125 a^{4} + 373 a^{3} + 183 a^{2} - 759 a - 265\) , \( 241 a^{4} - 2229 a^{3} + 3233 a^{2} + 1399 a + 390\bigr] \)
${y}^2+\left(-a^{4}+a^{3}+5a^{2}-3a-4\right){x}{y}+\left(a^{4}-a^{3}-4a^{2}+3a+2\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-125a^{4}+373a^{3}+183a^{2}-759a-265\right){x}+241a^{4}-2229a^{3}+3233a^{2}+1399a+390$
45.1-b2
45.1-b
$4$
$15$
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
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$3, 5$
3B.1.1 , 5B.4.2
$25$
\( 1 \)
$1$
$134.1402306$
1.45417284
\( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \)
\( \bigl[a\) , \( a^{4} - 2 a^{3} - 4 a^{2} + 7 a + 3\) , \( -a^{4} + 2 a^{3} + 5 a^{2} - 7 a - 4\) , \( 330 a^{4} + 154 a^{3} - 1529 a^{2} - 1577 a - 314\) , \( 9329 a^{4} + 3425 a^{3} - 41530 a^{2} - 38029 a - 6558\bigr] \)
${y}^2+a{x}{y}+\left(-a^{4}+2a^{3}+5a^{2}-7a-4\right){y}={x}^{3}+\left(a^{4}-2a^{3}-4a^{2}+7a+3\right){x}^{2}+\left(330a^{4}+154a^{3}-1529a^{2}-1577a-314\right){x}+9329a^{4}+3425a^{3}-41530a^{2}-38029a-6558$
225.1-a2
225.1-a
$4$
$15$
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
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
$3, 5$
3B , 5B.4.2
$25$
\( 2 \)
$1$
$9.391251630$
1.83253789
\( -\frac{660212871515640604}{5} a^{4} + \frac{1794053534107468711}{5} a^{3} + \frac{220022592976562427}{5} a^{2} - \frac{1698345910262749894}{5} a - \frac{384439131601574358}{5} \)
\( \bigl[a^{3} - 3 a - 1\) , \( -a^{3} + 5 a + 2\) , \( -a^{4} + a^{3} + 5 a^{2} - 2 a - 4\) , \( -5 a^{4} + 304 a^{3} - 129 a^{2} - 1151 a - 408\) , \( -3090 a^{4} + 1758 a^{3} + 12427 a^{2} + 1570 a - 1030\bigr] \)
${y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(-a^{4}+a^{3}+5a^{2}-2a-4\right){y}={x}^{3}+\left(-a^{3}+5a+2\right){x}^{2}+\left(-5a^{4}+304a^{3}-129a^{2}-1151a-408\right){x}-3090a^{4}+1758a^{3}+12427a^{2}+1570a-1030$
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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.