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-a6
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
\( \frac{22825881}{125} a + \frac{41720589}{125} \)
\( \bigl[a + 1\) , \( a + 1\) , \( a + 1\) , \( 4 a - 1\) , \( 9\bigr] \)
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(4a-1\right){x}+9$
25.1-b6
25.1-b
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
25.1
\( 5^{2} \)
\( 5^{4} \)
$0.91566$
$(-a), (-a+1)$
0
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2, 3$
2B , 3B
$1$
\( 3 \)
$1$
$31.19634696$
1.276425190
\( \frac{22825881}{125} a + \frac{41720589}{125} \)
\( \bigl[a\) , \( a\) , \( a\) , \( -3 a - 7\) , \( a + 1\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}+a{x}^{2}+\left(-3a-7\right){x}+a+1$
225.1-c6
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
$\Z/6\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B.1.1
$1$
\( 2 \cdot 3 \)
$1$
$24.88225636$
0.904958914
\( \frac{22825881}{125} a + \frac{41720589}{125} \)
\( \bigl[a\) , \( -a + 1\) , \( a + 1\) , \( 23 a - 70\) , \( -60 a + 164\bigr] \)
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(23a-70\right){x}-60a+164$
225.1-d6
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
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B.1.2
$1$
\( 2 \)
$1$
$8.294085455$
0.904958914
\( \frac{22825881}{125} a + \frac{41720589}{125} \)
\( \bigl[1\) , \( -1\) , \( a + 1\) , \( -2 a - 8\) , \( -3 a - 8\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-2a-8\right){x}-3a-8$
625.1-k6
625.1-k
$8$
$12$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
625.1
\( 5^{4} \)
\( 5^{16} \)
$2.04747$
$(-a), (-a+1)$
0
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2, 3$
2B , 3B
$1$
\( 2^{4} \)
$1$
$6.239269393$
1.361520203
\( \frac{22825881}{125} a + \frac{41720589}{125} \)
\( \bigl[a\) , \( -1\) , \( a\) , \( -102 a - 193\) , \( 835 a + 1487\bigr] \)
${y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+\left(-102a-193\right){x}+835a+1487$
625.1-l6
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
\( \frac{22825881}{125} a + \frac{41720589}{125} \)
\( \bigl[a + 1\) , \( -a - 1\) , \( a\) , \( 38 a - 132\) , \( -198 a + 516\bigr] \)
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(38a-132\right){x}-198a+516$
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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.