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
29241.1-CMe1
29241.1-CMe
$1$
$1$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
29241.1
\( 3^{4} \cdot 19^{2} \)
\( 3^{12} \cdot 19^{10} \)
$2.02394$
$(-2a+1), (-5a+3)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$19$
19Cs.4.1
$1$
\( 1 \)
$1$
$0.402493399$
0.464759345
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 2357 a - 1283\bigr] \)
${y}^2+a{y}={x}^{3}+2357a-1283$
29241.1-CMd1
29241.1-CMd
$2$
$3$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
29241.1
\( 3^{4} \cdot 19^{2} \)
\( 3^{12} \cdot 19^{6} \)
$2.02394$
$(-2a+1), (-5a+3)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$3, 19$
3Cs[2] , 19Cs.4.1
$1$
\( 1 \)
$1$
$1.074014050$
1.240164602
\( 0 \)
\( \bigl[0\) , \( 0\) , \( 1\) , \( 0\) , \( 42 a + 80\bigr] \)
${y}^2+{y}={x}^{3}+42a+80$
29241.1-CMd2
29241.1-CMd
$2$
$3$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
29241.1
\( 3^{4} \cdot 19^{2} \)
\( 3^{4} \cdot 19^{6} \)
$2.02394$
$(-2a+1), (-5a+3)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-27$
$\mathrm{U}(1)$
✓
✓
$19$
19Cs.4.1
$1$
\( 1 \)
$1$
$1.074014050$
1.240164602
\( -12288000 \)
\( \bigl[0\) , \( a + 1\) , \( 1\) , \( -49 a - 160\) , \( -480 a - 789\bigr] \)
${y}^2+{y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-49a-160\right){x}-480a-789$
29241.1-CMc1
29241.1-CMc
$1$
$1$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
29241.1
\( 3^{4} \cdot 19^{2} \)
\( 3^{12} \cdot 19^{2} \)
$2.02394$
$(-2a+1), (-5a+3)$
0
$\mathsf{trivial}$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$19$
19Cs.4.1
$1$
\( 1 \)
$1$
$2.865900863$
3.309257270
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( -6 a + 6\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}-6a+6$
29241.1-CMb1
29241.1-CMb
$1$
$1$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
29241.1
\( 3^{4} \cdot 19^{2} \)
\( 3^{6} \cdot 19^{8} \)
$2.02394$
$(-2a+1), (-5a+3)$
0
$\Z/3\Z$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$3$
3B.1.1[2]
$1$
\( 3^{2} \)
$1$
$1.138793527$
1.314965499
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a + 1\) , \( 0\) , \( 57 a - 104\bigr] \)
${y}^2+\left(a+1\right){y}={x}^{3}+57a-104$
29241.1-CMa1
29241.1-CMa
$1$
$1$
\(\Q(\sqrt{-3}) \)
$2$
$[0, 1]$
29241.1
\( 3^{4} \cdot 19^{2} \)
\( 3^{6} \cdot 19^{4} \)
$2.02394$
$(-2a+1), (-5a+3)$
0
$\Z/3\Z$
$\textsf{yes}$
$-3$
$\mathrm{U}(1)$
✓
✓
$3$
3B.1.1[2]
$1$
\( 3^{2} \)
$1$
$3.038758527$
3.508856107
\( 0 \)
\( \bigl[0\) , \( 0\) , \( a\) , \( 0\) , \( 5 a - 1\bigr] \)
${y}^2+a{y}={x}^{3}+5a-1$
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.