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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1680.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1680.a1 | 1680l3 | \([0, -1, 0, -301, 1960]\) | \(189123395584/16078125\) | \(257250000\) | \([2]\) | \(864\) | \(0.35520\) | |
1680.a2 | 1680l1 | \([0, -1, 0, -61, -164]\) | \(1594753024/4725\) | \(75600\) | \([2]\) | \(288\) | \(-0.19410\) | \(\Gamma_0(N)\)-optimal |
1680.a3 | 1680l2 | \([0, -1, 0, -36, -324]\) | \(-20720464/178605\) | \(-45722880\) | \([2]\) | \(576\) | \(0.15247\) | |
1680.a4 | 1680l4 | \([0, -1, 0, 324, 8460]\) | \(14647977776/132355125\) | \(-33882912000\) | \([2]\) | \(1728\) | \(0.70178\) |
Rank
sage: E.rank()
The elliptic curves in class 1680.a have rank \(0\).
Complex multiplication
The elliptic curves in class 1680.a do not have complex multiplication.Modular form 1680.2.a.a
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.