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
28.1-a1
28.1-a
$1$
$1$
\(\Q(\sqrt{53}) \)
$2$
$[2, 0]$
28.1
\( 2^{2} \cdot 7 \)
\( 2^{2} \cdot 7^{6} \)
$1.49646$
$(-a-2), (2)$
0
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 2 \)
$1$
$4.778422636$
1.312733656
\( -\frac{6206419620}{117649} a - \frac{38967954713}{235298} \)
\( \bigl[a\) , \( -1\) , \( 1\) , \( -10 a - 29\) , \( -43 a - 135\bigr] \)
${y}^2+a{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-10a-29\right){x}-43a-135$
28.1-b1
28.1-b
$1$
$1$
\(\Q(\sqrt{53}) \)
$2$
$[2, 0]$
28.1
\( 2^{2} \cdot 7 \)
\( 2^{4} \cdot 7^{4} \)
$1.49646$
$(-a-2), (2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 2^{2} \)
$0.095218482$
$22.21799898$
2.324760677
\( \frac{12774075}{9604} a - \frac{50451713}{9604} \)
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( -2 a\) , \( -a + 1\bigr] \)
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}-2a{x}-a+1$
28.1-c1
28.1-c
$2$
$2$
\(\Q(\sqrt{53}) \)
$2$
$[2, 0]$
28.1
\( 2^{2} \cdot 7 \)
\( - 2^{8} \cdot 7^{3} \)
$1.49646$
$(-a-2), (2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2 \)
$1$
$19.42800747$
1.334321031
\( -\frac{3528949}{686} a - \frac{20337669}{5488} \)
\( \bigl[a + 1\) , \( a\) , \( a\) , \( 3 a + 6\) , \( 3 a + 7\bigr] \)
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(3a+6\right){x}+3a+7$
28.1-c2
28.1-c
$2$
$2$
\(\Q(\sqrt{53}) \)
$2$
$[2, 0]$
28.1
\( 2^{2} \cdot 7 \)
\( - 2^{4} \cdot 7^{6} \)
$1.49646$
$(-a-2), (2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{2} \)
$1$
$9.714003739$
1.334321031
\( \frac{21413443562485}{235298} a + \frac{134478824713501}{470596} \)
\( \bigl[a + 1\) , \( a\) , \( a\) , \( -17 a - 54\) , \( 31 a + 91\bigr] \)
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-17a-54\right){x}+31a+91$
28.1-d1
28.1-d
$1$
$1$
\(\Q(\sqrt{53}) \)
$2$
$[2, 0]$
28.1
\( 2^{2} \cdot 7 \)
\( 2^{6} \cdot 7^{2} \)
$1.49646$
$(-a-2), (2)$
$1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$1$
\( 2 \)
$0.397928512$
$9.608278544$
2.100741912
\( -\frac{109125}{392} a + \frac{334375}{392} \)
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( 4 a + 14\) , \( -5 a - 16\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+14\right){x}-5a-16$
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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.