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
512.1-a1 512.1-a $1$
$1$
3.3.361.1 $3$
$[3, 0]$
512.1
\( 2^{9} \)
\( - 2^{33} \)
$4.80217$
$(2)$
$0 \le r \le 1$
$\mathsf{trivial}$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
\( 1 \)
$1$
$6.133714023$
4.445419974
\( -722 \)
\( \bigl[0\) , \( a + 1\) , \( 0\) , \( -18 a^{2} - 23 a + 56\) , \( -147 a^{2} - 189 a + 450\bigr] \) ${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-18a^{2}-23a+56\right){x}-147a^{2}-189a+450$
512.1-b1 512.1-b $4$
$4$
3.3.361.1 $3$
$[3, 0]$
512.1
\( 2^{9} \)
\( - 2^{24} \)
$4.80217$
$(2)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
$2, 5$
2B , 5S4 $1$
\( 2^{2} \)
$0.430691078$
$48.05858603$
3.268169091
\( -539775685804387088 a^{2} - 119763357649243392 a + 3092318123002094048 \)
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 2457 a^{2} + 497 a - 14194\) , \( -95805 a^{2} - 20965 a + 549589\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(2457a^{2}+497a-14194\right){x}-95805a^{2}-20965a+549589$
512.1-b2 512.1-b $4$
$4$
3.3.361.1 $3$
$[3, 0]$
512.1
\( 2^{9} \)
\( - 2^{24} \)
$4.80217$
$(2)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
$2, 5$
2B , 5S4 $1$
\( 2^{2} \)
$0.430691078$
$48.05858603$
3.268169091
\( 119763357649243392 a^{2} - 420012328155143696 a + 334398591538328736 \)
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 112 a^{2} - 18 a - 769\) , \( -1208 a^{2} + 27 a + 7679\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(112a^{2}-18a-769\right){x}-1208a^{2}+27a+7679$
512.1-b3 512.1-b $4$
$4$
3.3.361.1 $3$
$[3, 0]$
512.1
\( 2^{9} \)
\( 2^{12} \)
$4.80217$
$(2)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
✓
$2, 5$
2Cs , 5S4 $1$
\( 2 \)
$0.861382156$
$192.2343441$
3.268169091
\( 1462911232 \)
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( 112 a^{2} - 23 a - 759\) , \( -1232 a^{2} + 22 a + 7799\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(112a^{2}-23a-759\right){x}-1232a^{2}+22a+7799$
512.1-b4 512.1-b $4$
$4$
3.3.361.1 $3$
$[3, 0]$
512.1
\( 2^{9} \)
\( - 2^{24} \)
$4.80217$
$(2)$
$1$
$\Z/2\Z$
$\textsf{no}$
$\mathrm{SU}(2)$ ✓
✓
$2, 5$
2B , 5S4 $1$
\( 2^{2} \)
$0.430691078$
$48.05858603$
3.268169091
\( 420012328155143696 a^{2} + 539775685804387088 a - 1286609618640416176 \)
\( \bigl[0\) , \( -a + 1\) , \( 0\) , \( -558 a^{2} - 883 a + 1296\) , \( 15339 a^{2} + 21339 a - 42910\bigr] \) ${y}^2={x}^{3}+\left(-a+1\right){x}^{2}+\left(-558a^{2}-883a+1296\right){x}+15339a^{2}+21339a-42910$
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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.
