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
25.2-a1
25.2-a
$4$
$14$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.2
\( 5^{2} \)
\( 5^{6} \)
$0.91566$
$(-a)$
0
$\Z/2\Z$
$\textsf{potential}$
$-7$
$N(\mathrm{U}(1))$
✓
✓
$1$
\( 2 \)
$1$
$11.70083348$
1.276665598
\( -3375 \)
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( -a - 3\) , \( -a - 1\bigr] \)
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a-3\right){x}-a-1$
25.2-a2
25.2-a
$4$
$14$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.2
\( 5^{2} \)
\( 5^{6} \)
$0.91566$
$(-a)$
0
$\Z/2\Z$
$\textsf{potential}$
$-7$
$N(\mathrm{U}(1))$
✓
✓
$1$
\( 2 \)
$1$
$11.70083348$
1.276665598
\( -3375 \)
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -3 a - 5\) , \( 4 a + 7\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-3a-5\right){x}+4a+7$
25.2-a3
25.2-a
$4$
$14$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.2
\( 5^{2} \)
\( 5^{6} \)
$0.91566$
$(-a)$
0
$\Z/2\Z$
$\textsf{potential}$
$-28$
$N(\mathrm{U}(1))$
✓
✓
$1$
\( 2 \)
$1$
$11.70083348$
1.276665598
\( 16581375 \)
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( 14 a - 53\) , \( -44 a + 109\bigr] \)
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(14a-53\right){x}-44a+109$
25.2-a4
25.2-a
$4$
$14$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.2
\( 5^{2} \)
\( 5^{6} \)
$0.91566$
$(-a)$
0
$\Z/2\Z$
$\textsf{potential}$
$-28$
$N(\mathrm{U}(1))$
✓
✓
$1$
\( 2 \)
$1$
$11.70083348$
1.276665598
\( 16581375 \)
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -43 a - 80\) , \( 245 a + 437\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-43a-80\right){x}+245a+437$
25.2-b1
25.2-b
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.2
\( 5^{2} \)
\( 5^{4} \)
$0.91566$
$(-a)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$7$
7Ns.2.1
$1$
\( 1 \)
$0.139669982$
$16.42661250$
1.001316654
\( 0 \)
\( \bigl[0\) , \( a + 1\) , \( a + 1\) , \( a + 2\) , \( -2 a + 4\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(a+2\right){x}-2a+4$
25.2-b2
25.2-b
$2$
$3$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.2
\( 5^{2} \)
\( 5^{4} \)
$0.91566$
$(-a)$
$1$
$\mathsf{trivial}$
$\textsf{potential}$
$-3$
$N(\mathrm{U}(1))$
✓
✓
$7$
7Ns.2.1
$1$
\( 3 \)
$0.046556660$
$16.42661250$
1.001316654
\( 0 \)
\( \bigl[0\) , \( -a - 1\) , \( a + 1\) , \( a + 2\) , \( -a - 2\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(a+2\right){x}-a-2$
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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.