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
67.2-a1
67.2-a
$1$
$1$
6.6.592661.1
$6$
$[6, 0]$
67.2
\( 67 \)
\( 67^{5} \)
$97.65953$
$(-a^4+a^3+5a^2-2a-3)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 5 \)
$0.020626064$
$3516.974403$
2.82685
\( -\frac{184392193330736}{1350125107} a^{5} + \frac{272761023393987}{1350125107} a^{4} + \frac{846288828316947}{1350125107} a^{3} - \frac{1135825837953888}{1350125107} a^{2} - \frac{622639562545579}{1350125107} a + \frac{611921133846794}{1350125107} \)
\( \bigl[a^{5} - a^{4} - 4 a^{3} + 5 a^{2} + 2 a - 4\) , \( a^{5} - 2 a^{4} - 5 a^{3} + 8 a^{2} + 5 a - 3\) , \( 2 a^{5} - a^{4} - 9 a^{3} + 4 a^{2} + 6 a - 3\) , \( 4 a^{5} - 3 a^{4} - 19 a^{3} + 11 a^{2} + 14 a - 6\) , \( a^{5} - a^{4} - 6 a^{3} + 3 a^{2} + 6 a - 3\bigr] \)
${y}^2+\left(a^{5}-a^{4}-4a^{3}+5a^{2}+2a-4\right){x}{y}+\left(2a^{5}-a^{4}-9a^{3}+4a^{2}+6a-3\right){y}={x}^{3}+\left(a^{5}-2a^{4}-5a^{3}+8a^{2}+5a-3\right){x}^{2}+\left(4a^{5}-3a^{4}-19a^{3}+11a^{2}+14a-6\right){x}+a^{5}-a^{4}-6a^{3}+3a^{2}+6a-3$
67.2-b1
67.2-b
$1$
$1$
6.6.592661.1
$6$
$[6, 0]$
67.2
\( 67 \)
\( 67 \)
$97.65953$
$(-a^4+a^3+5a^2-2a-3)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 1 \)
$0.002280473$
$122461.4129$
2.17657
\( -\frac{75626}{67} a^{5} + \frac{180467}{67} a^{4} + \frac{112536}{67} a^{3} - \frac{337206}{67} a^{2} - \frac{114765}{67} a + \frac{164450}{67} \)
\( \bigl[2 a^{5} - a^{4} - 9 a^{3} + 5 a^{2} + 5 a - 5\) , \( 2 a^{5} - 2 a^{4} - 9 a^{3} + 7 a^{2} + 6 a - 2\) , \( a^{2} - 1\) , \( -4 a^{5} + 17 a^{3} - a^{2} - 6 a + 6\) , \( 11 a^{5} - 4 a^{4} - 60 a^{3} + 5 a^{2} + 70 a + 26\bigr] \)
${y}^2+\left(2a^{5}-a^{4}-9a^{3}+5a^{2}+5a-5\right){x}{y}+\left(a^{2}-1\right){y}={x}^{3}+\left(2a^{5}-2a^{4}-9a^{3}+7a^{2}+6a-2\right){x}^{2}+\left(-4a^{5}+17a^{3}-a^{2}-6a+6\right){x}+11a^{5}-4a^{4}-60a^{3}+5a^{2}+70a+26$
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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.