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-a1
55.1-a
$2$
$2$
4.4.5125.1
$4$
$[4, 0]$
55.1
\( 5 \cdot 11 \)
\( 5^{2} \cdot 11^{2} \)
$10.55676$
$(-a^3+a^2+4a+1), (-a)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{2} \)
$1$
$134.2078364$
1.874696378
\( -\frac{287491768525988172527}{605} a^{3} + \frac{1065548612442610734672}{605} a^{2} - \frac{93261222637284540808}{605} a - \frac{370660991375398969213}{121} \)
\( \bigl[a^{3} - a^{2} - 3 a + 1\) , \( -a^{3} + a^{2} + 3 a - 1\) , \( a + 1\) , \( -66 a^{3} + 194 a^{2} + 198 a - 666\) , \( -655 a^{3} + 2069 a^{2} + 1528 a - 6306\bigr] \)
${y}^2+\left(a^{3}-a^{2}-3a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+3a-1\right){x}^{2}+\left(-66a^{3}+194a^{2}+198a-666\right){x}-655a^{3}+2069a^{2}+1528a-6306$
55.1-f1
55.1-f
$2$
$2$
4.4.5125.1
$4$
$[4, 0]$
55.1
\( 5 \cdot 11 \)
\( 5^{2} \cdot 11^{2} \)
$10.55676$
$(-a^3+a^2+4a+1), (-a)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{2} \)
$1$
$70.55402936$
0.985541431
\( -\frac{287491768525988172527}{605} a^{3} + \frac{1065548612442610734672}{605} a^{2} - \frac{93261222637284540808}{605} a - \frac{370660991375398969213}{121} \)
\( \bigl[a^{3} - 4 a - 4\) , \( 1\) , \( 1\) , \( -218 a^{3} - 146 a^{2} + 869 a + 827\) , \( -4041 a^{3} - 2859 a^{2} + 16568 a + 16518\bigr] \)
${y}^2+\left(a^{3}-4a-4\right){x}{y}+{y}={x}^{3}+{x}^{2}+\left(-218a^{3}-146a^{2}+869a+827\right){x}-4041a^{3}-2859a^{2}+16568a+16518$
275.1-f2
275.1-f
$2$
$2$
4.4.5125.1
$4$
$[4, 0]$
275.1
\( 5^{2} \cdot 11 \)
\( 5^{8} \cdot 11^{2} \)
$12.90928$
$(-a^3+a^2+4a+1), (-a)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2B
$1$
\( 2^{3} \)
$2.630873448$
$13.28471173$
3.905666397
\( -\frac{287491768525988172527}{605} a^{3} + \frac{1065548612442610734672}{605} a^{2} - \frac{93261222637284540808}{605} a - \frac{370660991375398969213}{121} \)
\( \bigl[a^{3} - 5 a - 3\) , \( -a^{2} + 3\) , \( a^{2} - 3\) , \( -191 a^{3} - 180 a^{2} + 655 a + 688\) , \( 5406 a^{3} + 3362 a^{2} - 23384 a - 22836\bigr] \)
${y}^2+\left(a^{3}-5a-3\right){x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-191a^{3}-180a^{2}+655a+688\right){x}+5406a^{3}+3362a^{2}-23384a-22836$
275.1-q2
275.1-q
$2$
$2$
4.4.5125.1
$4$
$[4, 0]$
275.1
\( 5^{2} \cdot 11 \)
\( 5^{8} \cdot 11^{2} \)
$12.90928$
$(-a^3+a^2+4a+1), (-a)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2B
$1$
\( 2^{3} \)
$0.216192880$
$142.5533925$
3.443990916
\( -\frac{287491768525988172527}{605} a^{3} + \frac{1065548612442610734672}{605} a^{2} - \frac{93261222637284540808}{605} a - \frac{370660991375398969213}{121} \)
\( \bigl[a\) , \( -a^{3} + 6 a + 3\) , \( a^{3} - 4 a - 4\) , \( 106 a^{3} - 409 a^{2} + 92 a + 629\) , \( -1849 a^{3} + 6803 a^{2} - 394 a - 12131\bigr] \)
${y}^2+a{x}{y}+\left(a^{3}-4a-4\right){y}={x}^{3}+\left(-a^{3}+6a+3\right){x}^{2}+\left(106a^{3}-409a^{2}+92a+629\right){x}-1849a^{3}+6803a^{2}-394a-12131$
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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.