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
8.1-a1 8.1-a $4$
$21$
\(\Q(\zeta_{9})^+\) $3$
$[3, 0]$
8.1
\( 2^{3} \)
\( - 2^{9} \)
$1.13736$
$(2)$
0
$\Z/21\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
$3, 7$
3B.1.1 , 7B.1.1 $1$
\( 3 \)
$1$
$386.8008344$
0.292366465
\( -\frac{140625}{8} \)
\( \bigl[a^{2} + a - 1\) , \( -a^{2} + a + 2\) , \( a^{2} - 2\) , \( 13 a^{2} + 2 a - 44\) , \( -22 a^{2} - 3 a + 88\bigr] \) ${y}^2+\left(a^{2}+a-1\right){x}{y}+\left(a^{2}-2\right){y}={x}^{3}+\left(-a^{2}+a+2\right){x}^{2}+\left(13a^{2}+2a-44\right){x}-22a^{2}-3a+88$
8.1-a2 8.1-a $4$
$21$
\(\Q(\zeta_{9})^+\) $3$
$[3, 0]$
8.1
\( 2^{3} \)
\( - 2^{63} \)
$1.13736$
$(2)$
0
$\Z/3\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
$3, 7$
3B.1.1 , 7B.1.3 $1$
\( 3 \cdot 7 \)
$1$
$1.127699225$
0.292366465
\( -\frac{1159088625}{2097152} \)
\( \bigl[a\) , \( -1\) , \( a^{2} - 1\) , \( 125040 a^{2} - 43189 a - 360486\) , \( 56242078 a^{2} - 19527401 a - 161952511\bigr] \) ${y}^2+a{x}{y}+\left(a^{2}-1\right){y}={x}^{3}-{x}^{2}+\left(125040a^{2}-43189a-360486\right){x}+56242078a^{2}-19527401a-161952511$
8.1-a3 8.1-a $4$
$21$
\(\Q(\zeta_{9})^+\) $3$
$[3, 0]$
8.1
\( 2^{3} \)
\( - 2^{3} \)
$1.13736$
$(2)$
0
$\Z/7\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
$3, 7$
3B.1.2 , 7B.1.1 $1$
\( 1 \)
$1$
$128.9336114$
0.292366465
\( \frac{3375}{2} \)
\( \bigl[1\) , \( a^{2} + a - 3\) , \( a^{2} + a - 1\) , \( -a^{2} + 4\) , \( -a - 2\bigr] \) ${y}^2+{x}{y}+\left(a^{2}+a-1\right){y}={x}^{3}+\left(a^{2}+a-3\right){x}^{2}+\left(-a^{2}+4\right){x}-a-2$
8.1-a4 8.1-a $4$
$21$
\(\Q(\zeta_{9})^+\) $3$
$[3, 0]$
8.1
\( 2^{3} \)
\( - 2^{21} \)
$1.13736$
$(2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
$3, 7$
3B.1.2 , 7B.1.3 $1$
\( 7 \)
$1$
$0.375899741$
0.292366465
\( -\frac{189613868625}{128} \)
\( \bigl[a + 1\) , \( -a\) , \( a^{2} - 1\) , \( 17115 a^{2} - 5866 a - 49431\) , \( 1410206 a^{2} - 489214 a - 4061553\bigr] \) ${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}-a{x}^{2}+\left(17115a^{2}-5866a-49431\right){x}+1410206a^{2}-489214a-4061553$
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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.
