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
448.1-c1
448.1-c
$4$
$4$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
448.1
\( 2^{6} \cdot 7 \)
\( 2^{16} \cdot 7^{4} \)
$1.88394$
$(a+3), (2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{2} \)
$1$
$5.784310009$
1.262239926
\( \frac{11664}{49} \)
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -2 a + 11\) , \( 13 a - 33\bigr] \)
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-2a+11\right){x}+13a-33$
448.1-f1
448.1-f
$4$
$4$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
448.1
\( 2^{6} \cdot 7 \)
\( 2^{16} \cdot 7^{4} \)
$1.88394$
$(a+3), (2)$
0
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{2} \)
$1$
$5.784310009$
1.262239926
\( \frac{11664}{49} \)
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( 4 a + 8\) , \( -16 a - 28\bigr] \)
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(4a+8\right){x}-16a-28$
1792.1-m1
1792.1-m
$4$
$4$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
1792.1
\( 2^{8} \cdot 7 \)
\( 2^{16} \cdot 7^{4} \)
$2.66430$
$(a+3), (2)$
$1$
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{4} \)
$1.593647072$
$5.784310009$
4.023129928
\( \frac{11664}{49} \)
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( 4 a + 8\) , \( 16 a + 28\bigr] \)
${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(4a+8\right){x}+16a+28$
1792.1-bd1
1792.1-bd
$4$
$4$
\(\Q(\sqrt{21}) \)
$2$
$[2, 0]$
1792.1
\( 2^{8} \cdot 7 \)
\( 2^{16} \cdot 7^{4} \)
$2.66430$
$(a+3), (2)$
$1$
$\Z/4\Z$
$\textsf{no}$
$\mathrm{SU}(2)$
✓
$2$
2B
$1$
\( 2^{4} \)
$1.593647072$
$5.784310009$
4.023129928
\( \frac{11664}{49} \)
\( \bigl[0\) , \( -a - 1\) , \( 0\) , \( -2 a + 11\) , \( -13 a + 33\bigr] \)
${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a+11\right){x}-13a+33$
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.