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
3.1-a2
3.1-a
$4$
$6$
\(\Q(\sqrt{85}) \)
$2$
$[2, 0]$
3.1
\( 3 \)
\( - 3^{24} \)
$1.08425$
$(3,a)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B
$4$
\( 2 \)
$1$
$6.295137819$
1.365607129
\( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \)
\( \bigl[a\) , \( a\) , \( 0\) , \( -3 a + 21\) , \( -190\bigr] \)
${y}^2+a{x}{y}={x}^{3}+a{x}^{2}+\left(-3a+21\right){x}-190$
3.1-b2
3.1-b
$4$
$6$
\(\Q(\sqrt{85}) \)
$2$
$[2, 0]$
3.1
\( 3 \)
\( - 3^{12} \)
$1.08425$
$(3,a)$
$1$
$\Z/6\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B.1.1
$1$
\( 2^{2} \cdot 3 \)
$2.715008527$
$6.295137819$
1.235878333
\( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \)
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -4 a - 19\) , \( -29 a - 121\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-4a-19\right){x}-29a-121$
81.1-a2
81.1-a
$4$
$6$
\(\Q(\sqrt{85}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( - 3^{24} \)
$2.47155$
$(3,a), (3,a+2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2, 3$
2B , 3B.1.2
$1$
\( 2^{3} \)
$1$
$6.148387050$
1.333772417
\( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \)
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -47 a - 188\) , \( 262 a + 1083\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(-47a-188\right){x}+262a+1083$
81.1-c2
81.1-c
$4$
$6$
\(\Q(\sqrt{85}) \)
$2$
$[2, 0]$
81.1
\( 3^{4} \)
\( - 3^{36} \)
$2.47155$
$(3,a), (3,a+2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2, 3$
2B , 3B
$1$
\( 2^{3} \)
$1$
$6.148387050$
1.333772417
\( \frac{25207270205}{531441} a + \frac{103715849860}{531441} \)
\( \bigl[a + 1\) , \( -1\) , \( 1\) , \( -265361 a - 1090571\) , \( 158705406 a + 652243070\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(-265361a-1090571\right){x}+158705406a+652243070$
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.