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
121.1-a1 121.1-a $1$
$1$
\(\Q(\sqrt{89}) \) $2$
$[2, 0]$
121.1
\( 11^{2} \)
\( 11^{8} \)
$2.79595$
$(-2a+11), (2a+9)$
$0 \le r \le 1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
\( 1 \)
$1$
$0.312701784$
5.271378555
\( -\frac{191033112726655746813}{19487171} a - \frac{805584834173365923466}{19487171} \)
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 12551301 a - 65479965\) , \( 134825991618 a - 703385926245\bigr] \) ${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(12551301a-65479965\right){x}+134825991618a-703385926245$
121.1-b1 121.1-b $1$
$1$
\(\Q(\sqrt{89}) \) $2$
$[2, 0]$
121.1
\( 11^{2} \)
\( 11^{8} \)
$2.79595$
$(-2a+11), (2a+9)$
$0 \le r \le 1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
\( 1 \)
$1$
$0.312701784$
5.271378555
\( \frac{191033112726655746813}{19487171} a - \frac{90601631536365606389}{1771561} \)
\( \bigl[a + 1\) , \( a - 1\) , \( a\) , \( -12551290 a - 52928676\) , \( -134878920294 a - 568783134441\bigr] \) ${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-12551290a-52928676\right){x}-134878920294a-568783134441$
121.1-c1 121.1-c $3$
$25$
\(\Q(\sqrt{89}) \) $2$
$[2, 0]$
121.1
\( 11^{2} \)
\( 11^{2} \)
$2.79595$
$(-2a+11), (2a+9)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
$5$
5B.1.2 $25$
\( 1 \)
$1$
$0.064435690$
0.170754237
\( -\frac{52893159101157376}{11} \)
\( \bigl[0\) , \( -1\) , \( 1\) , \( -7820\) , \( -263580\bigr] \) ${y}^2+{y}={x}^{3}-{x}^{2}-7820{x}-263580$
121.1-c2 121.1-c $3$
$25$
\(\Q(\sqrt{89}) \) $2$
$[2, 0]$
121.1
\( 11^{2} \)
\( 11^{10} \)
$2.79595$
$(-2a+11), (2a+9)$
0
$\Z/5\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
$5$
5Cs.1.1 $1$
\( 5^{2} \)
$1$
$1.610892258$
0.170754237
\( -\frac{122023936}{161051} \)
\( \bigl[0\) , \( -1\) , \( 1\) , \( -10\) , \( -20\bigr] \) ${y}^2+{y}={x}^{3}-{x}^{2}-10{x}-20$
121.1-c3 121.1-c $3$
$25$
\(\Q(\sqrt{89}) \) $2$
$[2, 0]$
121.1
\( 11^{2} \)
\( 11^{2} \)
$2.79595$
$(-2a+11), (2a+9)$
0
$\Z/5\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
$5$
5B.1.1 $1$
\( 1 \)
$1$
$40.27230645$
0.170754237
\( -\frac{4096}{11} \)
\( \bigl[0\) , \( -1\) , \( 1\) , \( 0\) , \( 0\bigr] \) ${y}^2+{y}={x}^{3}-{x}^{2}$
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Pari/GP
SageMath
Magma
Oscar
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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.
