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
15.1-a3
15.1-a
$6$
$8$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
15.1
\( 3 \cdot 5 \)
\( 3^{2} \cdot 5^{8} \)
$0.80588$
$(-a+2), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2^{2} \)
$1$
$16.31232646$
0.889910366
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \)
\( \bigl[1\) , \( -a\) , \( a + 1\) , \( -41 a - 75\) , \( 73 a + 131\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-41a-75\right){x}+73a+131$
15.1-b3
15.1-b
$6$
$8$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
15.1
\( 3 \cdot 5 \)
\( 3^{2} \cdot 5^{8} \)
$0.80588$
$(-a+2), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2^{4} \)
$1$
$4.882598147$
1.065470266
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \)
\( \bigl[a\) , \( -a - 1\) , \( 0\) , \( 15 a - 49\) , \( 66 a - 181\bigr] \)
${y}^2+a{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(15a-49\right){x}+66a-181$
45.2-a3
45.2-a
$6$
$8$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
45.2
\( 3^{2} \cdot 5 \)
\( 3^{8} \cdot 5^{8} \)
$1.06060$
$(-a+2), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2Cs
$1$
\( 2^{5} \)
$1$
$2.955229559$
1.289767919
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \)
\( \bigl[1\) , \( -a + 1\) , \( 1\) , \( 239 a - 672\) , \( 3320 a - 9264\bigr] \)
${y}^2+{x}{y}+{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(239a-672\right){x}+3320a-9264$
45.2-b3
45.2-b
$6$
$8$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
45.2
\( 3^{2} \cdot 5 \)
\( 3^{8} \cdot 5^{8} \)
$1.06060$
$(-a+2), (-a+1)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2Cs
$1$
\( 2^{3} \)
$0.641947094$
$8.983682810$
1.258473281
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \)
\( \bigl[a + 1\) , \( a + 1\) , \( 1\) , \( -12 a - 69\) , \( -110 a - 59\bigr] \)
${y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-12a-69\right){x}-110a-59$
1875.1-k3
1875.1-k
$6$
$8$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
1875.1
\( 3 \cdot 5^{4} \)
\( 3^{2} \cdot 5^{20} \)
$2.69463$
$(-a+2), (-a), (-a+1)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2Cs
$1$
\( 2^{5} \)
$1.940369156$
$3.262465293$
5.525614808
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \)
\( \bigl[1\) , \( a + 1\) , \( a + 1\) , \( -1016 a - 1903\) , \( 3401 a + 6401\bigr] \)
${y}^2+{x}{y}+\left(a+1\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-1016a-1903\right){x}+3401a+6401$
1875.1-bj3
1875.1-bj
$6$
$8$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
1875.1
\( 3 \cdot 5^{4} \)
\( 3^{2} \cdot 5^{20} \)
$2.69463$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
$2$
2Cs
$4$
\( 2^{5} \)
$1$
$0.976519629$
1.704752426
\( -\frac{127041323975657}{1171875} a + \frac{70922198887903}{234375} \)
\( \bigl[a\) , \( -a\) , \( a + 1\) , \( 372 a - 1250\) , \( 7384 a - 20759\bigr] \)
${y}^2+a{x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(372a-1250\right){x}+7384a-20759$
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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.