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
75.1-a4
75.1-a
$8$
$16$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
75.1
\( 3 \cdot 5^{2} \)
\( 3^{8} \cdot 5^{8} \)
$1.20507$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$2$
2Cs
$1$
\( 2^{5} \)
$1$
$7.846755528$
0.856151218
\( \frac{111284641}{50625} \)
\( \bigl[1\) , \( 1\) , \( 1\) , \( -10\) , \( -10\bigr] \)
${y}^2+{x}{y}+{y}={x}^{3}+{x}^{2}-10{x}-10$
75.1-b4
75.1-b
$8$
$16$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
75.1
\( 3 \cdot 5^{2} \)
\( 3^{8} \cdot 5^{8} \)
$1.20507$
$(-a+2), (-a), (-a+1)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
✓
$2$
2Cs
$1$
\( 2^{5} \)
$0.172294519$
$10.19195692$
1.532778450
\( \frac{111284641}{50625} \)
\( \bigl[a\) , \( a + 1\) , \( 1\) , \( 52 a - 136\) , \( -134 a + 380\bigr] \)
${y}^2+a{x}{y}+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(52a-136\right){x}-134a+380$
225.1-a4
225.1-a
$8$
$16$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( 3^{14} \cdot 5^{8} \)
$1.58597$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2^{5} \)
$1$
$5.163131942$
2.253375518
\( \frac{111284641}{50625} \)
\( \bigl[a + 1\) , \( a + 1\) , \( 0\) , \( 34 a - 83\) , \( 61 a - 168\bigr] \)
${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(34a-83\right){x}+61a-168$
225.1-b4
225.1-b
$8$
$16$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
225.1
\( 3^{2} \cdot 5^{2} \)
\( 3^{14} \cdot 5^{8} \)
$1.58597$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2Cs
$1$
\( 2^{5} \)
$1$
$5.163131942$
2.253375518
\( \frac{111284641}{50625} \)
\( \bigl[a\) , \( a\) , \( 1\) , \( -29 a - 57\) , \( -117 a - 205\bigr] \)
${y}^2+a{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-29a-57\right){x}-117a-205$
1875.1-u4
1875.1-u
$8$
$16$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
1875.1
\( 3 \cdot 5^{4} \)
\( 3^{8} \cdot 5^{20} \)
$2.69463$
$(-a+2), (-a), (-a+1)$
0
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2$
2Cs
$1$
\( 2^{5} \)
$1$
$2.038391385$
0.889626935
\( \frac{111284641}{50625} \)
\( \bigl[a + 1\) , \( -a + 1\) , \( a\) , \( -1254 a - 2255\) , \( 16189 a + 28995\bigr] \)
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-1254a-2255\right){x}+16189a+28995$
1875.1-x4
1875.1-x
$8$
$16$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
1875.1
\( 3 \cdot 5^{4} \)
\( 3^{8} \cdot 5^{20} \)
$2.69463$
$(-a+2), (-a), (-a+1)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
✓
$2$
2Cs
$1$
\( 2^{7} \)
$0.654479105$
$1.569351105$
3.586131735
\( \frac{111284641}{50625} \)
\( \bigl[1\) , \( 0\) , \( 1\) , \( -251\) , \( -727\bigr] \)
${y}^2+{x}{y}+{y}={x}^{3}-251{x}-727$
Download
displayed columns for
results
to
Text
Pari/GP
SageMath
Magma
Oscar
CSV
*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.