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
31.1-a1 31.1-a $2$
$2$
3.3.169.1 $3$
$[3, 0]$
31.1
\( 31 \)
\( - 31^{2} \)
$2.05893$
$(2a^2-3a-3)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$2$
2B $1$
\( 2 \)
$1$
$27.63718754$
1.062968751
\( \frac{2423516191674}{961} a^{2} - \frac{5731917800790}{961} a - \frac{1751721853153}{961} \)
\( \bigl[a + 1\) , \( a^{2} - a - 3\) , \( a^{2} - 3\) , \( -4 a^{2} - 7 a - 5\) , \( 7 a^{2} + 7 a - 6\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(-4a^{2}-7a-5\right){x}+7a^{2}+7a-6$
31.1-a2 31.1-a $2$
$2$
3.3.169.1 $3$
$[3, 0]$
31.1
\( 31 \)
\( -31 \)
$2.05893$
$(2a^2-3a-3)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$2$
2B $1$
\( 1 \)
$1$
$55.27437509$
1.062968751
\( -\frac{32365068}{31} a^{2} + \frac{42115872}{31} a + \frac{116134499}{31} \)
\( \bigl[a + 1\) , \( a^{2} - a - 3\) , \( a^{2} - 3\) , \( a^{2} - 2 a - 5\) , \( a^{2} - 2 a - 5\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-3\right){x}^{2}+\left(a^{2}-2a-5\right){x}+a^{2}-2a-5$
31.1-b1 31.1-b $2$
$5$
3.3.169.1 $3$
$[3, 0]$
31.1
\( 31 \)
\( - 31^{10} \)
$2.05893$
$(2a^2-3a-3)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$5$
5B.1.2 $1$
\( 2 \cdot 5 \)
$1$
$1.439391393$
1.107224148
\( \frac{2034273426504158658495}{819628286980801} a^{2} - \frac{2593062233763744426365}{819628286980801} a - \frac{7428286237785028686301}{819628286980801} \)
\( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a\) , \( 668 a^{2} - 867 a - 2461\) , \( 12546 a^{2} - 16025 a - 45866\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(668a^{2}-867a-2461\right){x}+12546a^{2}-16025a-45866$
31.1-b2 31.1-b $2$
$5$
3.3.169.1 $3$
$[3, 0]$
31.1
\( 31 \)
\( - 31^{2} \)
$2.05893$
$(2a^2-3a-3)$
0
$\Z/5\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$5$
5B.1.1 $1$
\( 2 \)
$1$
$179.9239241$
1.107224148
\( -\frac{385258025}{961} a^{2} + \frac{915269800}{961} a + \frac{281241504}{961} \)
\( \bigl[a + 1\) , \( -a^{2} + 2\) , \( a\) , \( -7 a^{2} + 8 a + 24\) , \( -4 a^{2} + 5 a + 14\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2\right){x}^{2}+\left(-7a^{2}+8a+24\right){x}-4a^{2}+5a+14$
31.1-c1 31.1-c $1$
$1$
3.3.169.1 $3$
$[3, 0]$
31.1
\( 31 \)
\( - 31^{2} \)
$2.05893$
$(2a^2-3a-3)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
$1$
\( 2 \)
$0.026338543$
$80.09660173$
0.973674366
\( \frac{126638766}{961} a^{2} - \frac{161358095}{961} a - \frac{462350752}{961} \)
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + a + 4\) , \( 1\) , \( -3 a^{2} + 3 a + 12\) , \( -3 a^{2} + 5 a + 7\bigr] \) ${y}^2+\left(a^{2}-a-3\right){x}{y}+{y}={x}^{3}+\left(-a^{2}+a+4\right){x}^{2}+\left(-3a^{2}+3a+12\right){x}-3a^{2}+5a+7$
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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.
