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-a2
25.1-a
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.1
\( 5^{2} \)
\( 5^{16} \)
$0.91566$
$(-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2, 3$
2B , 3B
$1$
\( 2^{2} \)
$1$
$4.961531894$
1.082695022
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \)
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 29 a - 66\) , \( -125 a + 369\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(29a-66\right){x}-125a+369$
25.1-b2
25.1-b
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.1
\( 5^{2} \)
\( 5^{16} \)
$0.91566$
$(-a), (-a+1)$
0
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2, 3$
2B , 3B
$1$
\( 2^{4} \cdot 3 \)
$1$
$1.949771685$
1.276425190
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \)
\( \bigl[a\) , \( a\) , \( a\) , \( 22 a + 33\) , \( 56 a + 96\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(22a+33\right){x}+56a+96$
225.1-c2
225.1-c
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( 3^{6} \cdot 5^{16} \)
$1.58597$
$(-a+2), (-a), (-a+1)$
0
$\Z/6\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B.1.1
$1$
\( 2^{4} \cdot 3 \)
$1$
$3.110282045$
0.904958914
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \)
\( \bigl[a + 1\) , \( 0\) , \( a + 1\) , \( 285 a + 507\) , \( 1611 a + 2888\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(285a+507\right){x}+1611a+2888$
225.1-d2
225.1-d
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( 3^{6} \cdot 5^{16} \)
$1.58597$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B.1.2
$1$
\( 2^{4} \)
$1$
$1.036760681$
0.904958914
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \)
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( 28 a - 23\) , \( 42 a - 188\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(28a-23\right){x}+42a-188$
625.1-k2
625.1-k
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
625.1
\( 5^{4} \)
\( 5^{28} \)
$2.04747$
$(-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B
$4$
\( 2^{4} \)
$1$
$0.389954337$
1.361520203
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \)
\( \bigl[a\) , \( -1\) , \( a\) , \( 523 a + 807\) , \( 3210 a + 5112\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(523a+807\right){x}+3210a+5112$
625.1-l2
625.1-l
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
625.1
\( 5^{4} \)
\( 5^{28} \)
$2.04747$
$(-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B
$1$
\( 2^{4} \)
$1$
$0.992306378$
0.866156017
\( \frac{359104782699}{244140625} a - \frac{619846590969}{244140625} \)
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 663 a - 1757\) , \( -15323 a + 42641\bigr] \)
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(663a-1757\right){x}-15323a+42641$
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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.