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
55.1-c1
55.1-c
$4$
$6$
4.4.5125.1
$4$
$[4, 0]$
55.1
\( 5 \cdot 11 \)
\( 5 \cdot 11 \)
$10.55676$
$(-a^3+a^2+4a+1), (-a)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B
$4$
\( 1 \)
$1$
$117.3818148$
1.639660313
\( -\frac{19204557701238}{55} a^{3} - \frac{2347403181723}{55} a^{2} + \frac{110246499123534}{55} a + \frac{99540191518811}{55} \)
\( \bigl[a^{2} - 4\) , \( -a\) , \( 1\) , \( 32 a^{3} - 33 a^{2} - 128 a - 53\) , \( 141 a^{3} - 137 a^{2} - 564 a - 261\bigr] \)
${y}^2+\left(a^{2}-4\right){x}{y}+{y}={x}^{3}-a{x}^{2}+\left(32a^{3}-33a^{2}-128a-53\right){x}+141a^{3}-137a^{2}-564a-261$
55.1-d2
55.1-d
$4$
$6$
4.4.5125.1
$4$
$[4, 0]$
55.1
\( 5 \cdot 11 \)
\( 5 \cdot 11 \)
$10.55676$
$(-a^3+a^2+4a+1), (-a)$
0
$\Z/6\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B.1.1
$4$
\( 1 \)
$1$
$669.0430099$
1.038399794
\( -\frac{19204557701238}{55} a^{3} - \frac{2347403181723}{55} a^{2} + \frac{110246499123534}{55} a + \frac{99540191518811}{55} \)
\( \bigl[a^{3} - 4 a - 3\) , \( 0\) , \( a^{3} - 4 a - 3\) , \( -4 a^{3} + 16 a - 3\) , \( -24 a^{3} - 24 a^{2} + 99 a + 130\bigr] \)
${y}^2+\left(a^{3}-4a-3\right){x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(-4a^{3}+16a-3\right){x}-24a^{3}-24a^{2}+99a+130$
275.1-g1
275.1-g
$4$
$6$
4.4.5125.1
$4$
$[4, 0]$
275.1
\( 5^{2} \cdot 11 \)
\( 5^{7} \cdot 11 \)
$12.90928$
$(-a^3+a^2+4a+1), (-a)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2, 3$
2B , 3B
$1$
\( 2^{2} \)
$0.321774786$
$211.4553706$
3.801752031
\( -\frac{19204557701238}{55} a^{3} - \frac{2347403181723}{55} a^{2} + \frac{110246499123534}{55} a + \frac{99540191518811}{55} \)
\( \bigl[a^{3} - 4 a - 4\) , \( -a^{3} + 2 a^{2} + 3 a - 3\) , \( a^{3} - 4 a - 4\) , \( 10 a^{3} - 42 a^{2} - 3 a + 61\) , \( 116 a^{2} - 172 a - 334\bigr] \)
${y}^2+\left(a^{3}-4a-4\right){x}{y}+\left(a^{3}-4a-4\right){y}={x}^{3}+\left(-a^{3}+2a^{2}+3a-3\right){x}^{2}+\left(10a^{3}-42a^{2}-3a+61\right){x}+116a^{2}-172a-334$
275.1-l4
275.1-l
$4$
$6$
4.4.5125.1
$4$
$[4, 0]$
275.1
\( 5^{2} \cdot 11 \)
\( 5^{7} \cdot 11 \)
$12.90928$
$(-a^3+a^2+4a+1), (-a)$
$0 \le r \le 1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2, 3$
2B , 3B
\( 2^{2} \)
$1$
$74.27901451$
3.977414736
\( -\frac{19204557701238}{55} a^{3} - \frac{2347403181723}{55} a^{2} + \frac{110246499123534}{55} a + \frac{99540191518811}{55} \)
\( \bigl[1\) , \( -a^{2} + 3\) , \( a^{3} - 4 a - 3\) , \( -68 a^{3} + 7 a^{2} + 258 a + 71\) , \( -477 a^{3} - 59 a^{2} + 1839 a + 923\bigr] \)
${y}^2+{x}{y}+\left(a^{3}-4a-3\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-68a^{3}+7a^{2}+258a+71\right){x}-477a^{3}-59a^{2}+1839a+923$
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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.