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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 312.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 312.f1 | 312c3 | \([0, 1, 0, -832, -9520]\) | \(62275269892/39\) | \(39936\) | \([2]\) | \(64\) | \(0.20313\) | |
| 312.f2 | 312c2 | \([0, 1, 0, -52, -160]\) | \(61918288/1521\) | \(389376\) | \([2, 2]\) | \(32\) | \(-0.14345\) | |
| 312.f3 | 312c1 | \([0, 1, 0, -7, 2]\) | \(2725888/1053\) | \(16848\) | \([4]\) | \(16\) | \(-0.49002\) | \(\Gamma_0(N)\)-optimal |
| 312.f4 | 312c4 | \([0, 1, 0, 8, -448]\) | \(48668/85683\) | \(-87739392\) | \([2]\) | \(64\) | \(0.20313\) |
Rank
sage: E.rank()
The elliptic curves in class 312.f have rank \(0\).
Complex multiplication
The elliptic curves in class 312.f do not have complex multiplication.Modular form 312.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
