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
65536.1-c1 65536.1-c $2$
$2$
\(\Q(\sqrt{-3}) \) $2$
$[0, 1]$
65536.1
\( 2^{16} \)
\( 2^{18} \)
$2.47639$
$(2)$
$1$
$\Z/2\Z$
$\textsf{potential}$
$-4$
$N(\mathrm{U}(1))$ ✓
✓
✓
$1$
\( 2 \)
$0.608709031$
$4.861490513$
3.417028151
\( 1728 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( 0\bigr] \) ${y}^2={x}^{3}-2{x}$
4096.1-CMd1 4096.1-CMd $1$
$1$
\(\Q(\sqrt{-1}) \) $2$
$[0, 1]$
4096.1
\( 2^{12} \)
\( 2^{18} \)
$1.42974$
$(a+1)$
$2$
$\Z/2\Z$
$\textsf{yes}$
$-4$
$\mathrm{U}(1)$ ✓
✓
✓
$1$
\( 2^{2} \)
$0.092631671$
$4.861490513$
1.801311967
\( 1728 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( 0\bigr] \) ${y}^2={x}^{3}-2{x}$
1024.1-b1 1024.1-b $4$
$4$
\(\Q(\sqrt{-2}) \) $2$
$[0, 1]$
1024.1
\( 2^{10} \)
\( 2^{18} \)
$1.42974$
$(a)$
$1$
$\Z/2\Z$
$\textsf{potential}$
$-4$
$N(\mathrm{U}(1))$ ✓
✓
✓
$1$
\( 2 \)
$0.608709031$
$4.861490513$
2.092493851
\( 1728 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( 0\bigr] \) ${y}^2={x}^{3}-2{x}$
1024.1-k4 1024.1-k $8$
$16$
\(\Q(\sqrt{2}) \) $2$
$[2, 0]$
1024.1
\( 2^{10} \)
\( 2^{18} \)
$1.42974$
$(a)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{potential}$
$-4$
$N(\mathrm{U}(1))$ ✓
✓
✓
$2$
2Cs $1$
\( 2^{2} \)
$0.608709031$
$19.44596205$
2.092493851
\( 1728 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( 0\bigr] \) ${y}^2={x}^{3}-2{x}$
4096.1-c1 4096.1-c $2$
$2$
\(\Q(\sqrt{3}) \) $2$
$[2, 0]$
4096.1
\( 2^{12} \)
\( 2^{18} \)
$2.47639$
$(a+1)$
$1$
$\Z/2\Z$
$\textsf{potential}$
$-4$
$N(\mathrm{U}(1))$ ✓
✓
✓
$1$
\( 2 \)
$0.608709031$
$19.44596205$
3.417028151
\( 1728 \)
\( \bigl[0\) , \( 0\) , \( 0\) , \( -2\) , \( 0\bigr] \) ${y}^2={x}^{3}-2{x}$
256.1-e8 256.1-e $8$
$16$
\(\Q(\zeta_{16})^+\) $4$
$[4, 0]$
256.1
\( 2^{8} \)
\( 2^{12} \)
$8.08786$
$(a)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{potential}$
$-4$
$N(\mathrm{U}(1))$ ✓
✓
✓
$2$
2Cs $1$
\( 2 \)
$0.152177257$
$1512.581761$
2.543159752
\( 1728 \)
\( \bigl[a^{3} - 2 a\) , \( a^{2} - 3\) , \( 0\) , \( 1\) , \( 0\bigr] \) ${y}^2+\left(a^{3}-2a\right){x}{y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+{x}$
32.1-a6 32.1-a $8$
$16$
\(\Q(\sqrt{4 + \sqrt{2}})\) $4$
$[4, 0]$
32.1
\( 2^{5} \)
\( 2^{12} \)
$16.50047$
$(a-2)$
$1$
$\Z/2\Z\oplus\Z/2\Z$
$\textsf{potential}$
$-4$
$N(\mathrm{U}(1))$ ✓
✓
✓
$2$
2Cs $1$
\( 2 \)
$0.608709031$
$1512.581761$
3.844896143
\( 1728 \)
\( \bigl[a^{3} - 4 a\) , \( a^{2} - 3\) , \( a^{3} - 4 a\) , \( 2 a^{2} - 8\) , \( a^{2} - 4\bigr] \) ${y}^2+\left(a^{3}-4a\right){x}{y}+\left(a^{3}-4a\right){y}={x}^{3}+\left(a^{2}-3\right){x}^{2}+\left(2a^{2}-8\right){x}+a^{2}-4$
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Pari/GP
SageMath
Magma
Oscar
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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.
