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
25.1-a3
25.1-a
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.1
\( 5^{2} \)
\( 5^{8} \)
$0.91566$
$(-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2, 3$
2Cs , 3B
$1$
\( 2^{2} \)
$1$
$19.84612757$
1.082695022
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \)
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 29 a - 71\) , \( -118 a + 338\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(29a-71\right){x}-118a+338$
25.1-b3
25.1-b
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.1
\( 5^{2} \)
\( 5^{8} \)
$0.91566$
$(-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2, 3$
2Cs , 3B
$1$
\( 2^{2} \cdot 3 \)
$1$
$7.799086741$
1.276425190
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \)
\( \bigl[a\) , \( a\) , \( a\) , \( -3 a - 12\) , \( -4 a - 11\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-3a-12\right){x}-4a-11$
225.1-c3
225.1-c
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( 3^{6} \cdot 5^{8} \)
$1.58597$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/6\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2Cs , 3B.1.1
$1$
\( 2^{4} \cdot 3 \)
$1$
$12.44112818$
0.904958914
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \)
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( -75 a - 138\) , \( 72 a + 131\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-75a-138\right){x}+72a+131$
225.1-d3
225.1-d
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( 3^{6} \cdot 5^{8} \)
$1.58597$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2Cs , 3B.1.2
$1$
\( 2^{4} \)
$1$
$4.147042727$
0.904958914
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \)
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 13 a - 53\) , \( 51 a - 149\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(13a-53\right){x}+51a-149$
625.1-k3
625.1-k
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
625.1
\( 5^{4} \)
\( 5^{20} \)
$2.04747$
$(-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2Cs , 3B
$4$
\( 2^{4} \)
$1$
$1.559817348$
1.361520203
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \)
\( \bigl[a\) , \( -1\) , \( a\) , \( -102 a - 318\) , \( 460 a + 237\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-102a-318\right){x}+460a+237$
625.1-l3
625.1-l
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
625.1
\( 5^{4} \)
\( 5^{20} \)
$2.04747$
$(-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2Cs , 3B
$1$
\( 2^{4} \)
$1$
$3.969225515$
0.866156017
\( -\frac{118077162021}{15625} a + \frac{329617083726}{15625} \)
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 663 a - 1882\) , \( -14073 a + 39266\bigr] \)
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(663a-1882\right){x}-14073a+39266$
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Pari/GP
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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.