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
4.1-a1
4.1-a
$6$
$45$
\(\Q(\sqrt{13}) \)
$2$
$[2, 0]$
4.1
\( 2^{2} \)
\( 2^{10} \)
$0.45564$
$(2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$3, 5$
3B , 5B.1.2
$1$
\( 1 \)
$1$
$1.142260539$
0.316806072
\( -\frac{1250637664527933}{32} a - \frac{1629300280935823}{32} \)
\( \bigl[1\) , \( 1\) , \( a\) , \( -29 a + 2\) , \( -52 a - 106\bigr] \)
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-29a+2\right){x}-52a-106$
4.1-a2
4.1-a
$6$
$45$
\(\Q(\sqrt{13}) \)
$2$
$[2, 0]$
4.1
\( 2^{2} \)
\( 2^{2} \)
$0.45564$
$(2)$
0
$\Z/5\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$3, 5$
3B , 5B.1.1
$1$
\( 1 \)
$1$
$28.55651349$
0.316806072
\( -\frac{461373}{2} a - \frac{601423}{2} \)
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( -2 a - 2\) , \( 0\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(-2a-2\right){x}$
4.1-a3
4.1-a
$6$
$45$
\(\Q(\sqrt{13}) \)
$2$
$[2, 0]$
4.1
\( 2^{2} \)
\( 2^{30} \)
$0.45564$
$(2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$3, 5$
3Cs , 5B.1.2
$1$
\( 1 \)
$1$
$1.142260539$
0.316806072
\( -\frac{1680914269}{32768} \)
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 75 a - 172\) , \( 507 a - 1170\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(75a-172\right){x}+507a-1170$
4.1-a4
4.1-a
$6$
$45$
\(\Q(\sqrt{13}) \)
$2$
$[2, 0]$
4.1
\( 2^{2} \)
\( 2^{2} \)
$0.45564$
$(2)$
0
$\Z/5\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$3, 5$
3B , 5B.1.1
$1$
\( 1 \)
$1$
$28.55651349$
0.316806072
\( \frac{461373}{2} a - 531398 \)
\( \bigl[1\) , \( 1\) , \( a\) , \( a - 3\) , \( -a + 1\bigr] \)
${y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(a-3\right){x}-a+1$
4.1-a5
4.1-a
$6$
$45$
\(\Q(\sqrt{13}) \)
$2$
$[2, 0]$
4.1
\( 2^{2} \)
\( 2^{6} \)
$0.45564$
$(2)$
0
$\Z/5\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$3, 5$
3Cs , 5B.1.1
$1$
\( 1 \)
$1$
$28.55651349$
0.316806072
\( \frac{1331}{8} \)
\( \bigl[a + 1\) , \( 1\) , \( 1\) , \( 3\) , \( -a + 4\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+{x}^{2}+3{x}-a+4$
4.1-a6
4.1-a
$6$
$45$
\(\Q(\sqrt{13}) \)
$2$
$[2, 0]$
4.1
\( 2^{2} \)
\( 2^{10} \)
$0.45564$
$(2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$3, 5$
3B , 5B.1.2
$1$
\( 1 \)
$1$
$1.142260539$
0.316806072
\( \frac{1250637664527933}{32} a - \frac{719984486365939}{8} \)
\( \bigl[1\) , \( 1\) , \( a + 1\) , \( 28 a - 27\) , \( 51 a - 158\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(28a-27\right){x}+51a-158$
Download
displayed columns for
results
to
Text
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.