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
12.1-a1
12.1-a
$2$
$2$
\(\Q(\sqrt{85}) \)
$2$
$[2, 0]$
12.1
\( 2^{2} \cdot 3 \)
\( - 2^{2} \cdot 3^{24} \)
$1.53336$
$(3,a), (2)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{2} \cdot 3 \)
$0.253138682$
$9.720342945$
1.601333870
\( -\frac{2087352478285}{531441} a + \frac{21331791431651}{1062882} \)
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -23 a - 92\) , \( -57847 a - 237737\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-23a-92\right){x}-57847a-237737$
12.1-a2
12.1-a
$2$
$2$
\(\Q(\sqrt{85}) \)
$2$
$[2, 0]$
12.1
\( 2^{2} \cdot 3 \)
\( 2^{4} \cdot 3^{18} \)
$1.53336$
$(3,a), (2)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{2} \cdot 3 \)
$0.126569341$
$19.44068589$
1.601333870
\( \frac{4133602}{729} a + \frac{87766517}{2916} \)
\( \bigl[a + 1\) , \( a - 1\) , \( 1\) , \( -673 a - 2762\) , \( -23395 a - 96149\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-673a-2762\right){x}-23395a-96149$
12.1-b1
12.1-b
$2$
$2$
\(\Q(\sqrt{85}) \)
$2$
$[2, 0]$
12.1
\( 2^{2} \cdot 3 \)
\( - 2^{2} \cdot 3^{12} \)
$1.53336$
$(3,a), (2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$4$
\( 2 \)
$1$
$9.720342945$
2.108638445
\( -\frac{2087352478285}{531441} a + \frac{21331791431651}{1062882} \)
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 14 a - 36\) , \( -24 a + 162\bigr] \)
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(14a-36\right){x}-24a+162$
12.1-b2
12.1-b
$2$
$2$
\(\Q(\sqrt{85}) \)
$2$
$[2, 0]$
12.1
\( 2^{2} \cdot 3 \)
\( 2^{4} \cdot 3^{6} \)
$1.53336$
$(3,a), (2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{2} \)
$1$
$19.44068589$
2.108638445
\( \frac{4133602}{729} a + \frac{87766517}{2916} \)
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 4 a + 14\) , \( 4 a + 16\bigr] \)
${y}^2+\left(a+1\right){x}{y}={x}^{3}+{x}^{2}+\left(4a+14\right){x}+4a+16$
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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.