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-a3 5.1-a $4$
$4$
4.4.7625.1 $4$
$[4, 0]$
5.1
\( 5 \)
\( 5^{3} \)
$9.54179$
$(-1/4a^3+1/4a^2+9/4a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$2, 3$
2B , 3Ns $1$
\( 1 \)
$1$
$371.9611438$
1.064921656
\( \frac{223782067836}{5} a^{3} - 105106588136 a^{2} - \frac{1305402638524}{5} a + \frac{2655351465237}{5} \)
\( \bigl[1\) , \( a - 1\) , \( \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{5}{4} a\) , \( -\frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{5}{4} a + 1\) , \( \frac{3}{4} a^{3} + \frac{1}{4} a^{2} - \frac{23}{4} a - 6\bigr] \) ${y}^2+{x}{y}+\left(\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{5}{4}a\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{5}{4}a+1\right){x}+\frac{3}{4}a^{3}+\frac{1}{4}a^{2}-\frac{23}{4}a-6$
5.1-b1 5.1-b $4$
$4$
4.4.7625.1 $4$
$[4, 0]$
5.1
\( 5 \)
\( 5^{3} \)
$9.54179$
$(-1/4a^3+1/4a^2+9/4a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$2, 3$
2B , 3Ns $1$
\( 1 \)
$1$
$371.9611438$
1.064921656
\( \frac{223782067836}{5} a^{3} - 105106588136 a^{2} - \frac{1305402638524}{5} a + \frac{2655351465237}{5} \)
\( \bigl[\frac{1}{4} a^{3} + \frac{3}{4} a^{2} - \frac{9}{4} a - 5\) , \( -\frac{1}{4} a^{3} - \frac{3}{4} a^{2} + \frac{13}{4} a + 6\) , \( a\) , \( -13 a^{3} + 14 a^{2} + 87 a - 41\) , \( -\frac{169}{4} a^{3} + \frac{289}{4} a^{2} + \frac{1049}{4} a - 312\bigr] \) ${y}^2+\left(\frac{1}{4}a^{3}+\frac{3}{4}a^{2}-\frac{9}{4}a-5\right){x}{y}+a{y}={x}^{3}+\left(-\frac{1}{4}a^{3}-\frac{3}{4}a^{2}+\frac{13}{4}a+6\right){x}^{2}+\left(-13a^{3}+14a^{2}+87a-41\right){x}-\frac{169}{4}a^{3}+\frac{289}{4}a^{2}+\frac{1049}{4}a-312$
25.1-b1 25.1-b $4$
$4$
4.4.7625.1 $4$
$[4, 0]$
25.1
\( 5^{2} \)
\( 5^{9} \)
$11.66812$
$(-1/4a^3+1/4a^2+9/4a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ $2, 3$
2B , 3Ns $1$
\( 2 \)
$1$
$584.1698134$
3.344946621
\( \frac{223782067836}{5} a^{3} - 105106588136 a^{2} - \frac{1305402638524}{5} a + \frac{2655351465237}{5} \)
\( \bigl[\frac{1}{4} a^{3} + \frac{3}{4} a^{2} - \frac{5}{4} a - 5\) , \( -\frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{9}{4} a + 1\) , \( a^{2} - 5\) , \( -\frac{3}{2} a^{3} + \frac{3}{2} a^{2} + \frac{29}{2} a - 3\) , \( -\frac{5}{4} a^{3} - \frac{19}{4} a^{2} + \frac{69}{4} a + 40\bigr] \) ${y}^2+\left(\frac{1}{4}a^{3}+\frac{3}{4}a^{2}-\frac{5}{4}a-5\right){x}{y}+\left(a^{2}-5\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{9}{4}a+1\right){x}^{2}+\left(-\frac{3}{2}a^{3}+\frac{3}{2}a^{2}+\frac{29}{2}a-3\right){x}-\frac{5}{4}a^{3}-\frac{19}{4}a^{2}+\frac{69}{4}a+40$
25.1-d2 25.1-d $4$
$4$
4.4.7625.1 $4$
$[4, 0]$
25.1
\( 5^{2} \)
\( 5^{9} \)
$11.66812$
$(-1/4a^3+1/4a^2+9/4a-2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ $2, 3$
2B , 3Ns $1$
\( 2 \)
$1$
$47.36810746$
0.271228994
\( \frac{223782067836}{5} a^{3} - 105106588136 a^{2} - \frac{1305402638524}{5} a + \frac{2655351465237}{5} \)
\( \bigl[a^{2} - 5\) , \( -\frac{1}{4} a^{3} + \frac{5}{4} a^{2} + \frac{5}{4} a - 6\) , \( \frac{1}{4} a^{3} + \frac{3}{4} a^{2} - \frac{5}{4} a - 5\) , \( -a^{3} - a^{2} + 6 a + 5\) , \( \frac{47}{4} a^{3} + \frac{21}{4} a^{2} - \frac{419}{4} a - 124\bigr] \) ${y}^2+\left(a^{2}-5\right){x}{y}+\left(\frac{1}{4}a^{3}+\frac{3}{4}a^{2}-\frac{5}{4}a-5\right){y}={x}^{3}+\left(-\frac{1}{4}a^{3}+\frac{5}{4}a^{2}+\frac{5}{4}a-6\right){x}^{2}+\left(-a^{3}-a^{2}+6a+5\right){x}+\frac{47}{4}a^{3}+\frac{21}{4}a^{2}-\frac{419}{4}a-124$
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Pari/GP
SageMath
Magma
Oscar
CSV
*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.
