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-a8
25.1-a
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.1
\( 5^{2} \)
\( 5^{4} \)
$0.91566$
$(-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2, 3$
2B , 3B
$1$
\( 1 \)
$1$
$19.84612757$
1.082695022
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \)
\( \bigl[a\) , \( a\) , \( a + 1\) , \( -426 a - 773\) , \( 6632 a + 11894\bigr] \)
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(-426a-773\right){x}+6632a+11894$
25.1-b8
25.1-b
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.1
\( 5^{2} \)
\( 5^{4} \)
$0.91566$
$(-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2, 3$
2B , 3B
$4$
\( 3 \)
$1$
$1.949771685$
1.276425190
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \)
\( \bigl[a + 1\) , \( a + 1\) , \( a\) , \( 32 a - 162\) , \( 150 a - 921\bigr] \)
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(32a-162\right){x}+150a-921$
225.1-c8
225.1-c
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( 3^{6} \cdot 5^{4} \)
$1.58597$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B.1.2
$4$
\( 2 \)
$1$
$2.073521363$
0.904958914
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \)
\( \bigl[1\) , \( -1\) , \( a\) , \( -239 a - 564\) , \( -3697 a - 5977\bigr] \)
${y}^2+{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-239a-564\right){x}-3697a-5977$
225.1-d8
225.1-d
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( 3^{6} \cdot 5^{4} \)
$1.58597$
$(-a+2), (-a), (-a+1)$
0
$\Z/6\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B.1.1
$4$
\( 2 \cdot 3 \)
$1$
$6.220564091$
0.904958914
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \)
\( \bigl[a\) , \( -a + 1\) , \( a\) , \( 673 a - 1936\) , \( 15542 a - 43236\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(673a-1936\right){x}+15542a-43236$
625.1-k8
625.1-k
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
625.1
\( 5^{4} \)
\( 5^{16} \)
$2.04747$
$(-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B
$4$
\( 2^{4} \)
$1$
$0.389954337$
1.361520203
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \)
\( \bigl[a + 1\) , \( -a - 1\) , \( a + 1\) , \( 725 a - 4170\) , \( 25789 a - 109928\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(725a-4170\right){x}+25789a-109928$
625.1-l8
625.1-l
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
625.1
\( 5^{4} \)
\( 5^{16} \)
$2.04747$
$(-a), (-a+1)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B
$1$
\( 2^{2} \)
$1$
$3.969225515$
0.866156017
\( \frac{32714515537919631}{125} a + \frac{11720222826515406}{25} \)
\( \bigl[a\) , \( -1\) , \( a + 1\) , \( -10665 a - 19343\) , \( 910322 a + 1632068\bigr] \)
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-10665a-19343\right){x}+910322a+1632068$
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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.