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SageMath
E = EllipticCurve("hi1")
E.isogeny_class()
Elliptic curves in class 91200.hi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
91200.hi1 | 91200hk4 | \([0, 1, 0, -993633, 380344863]\) | \(26487576322129/44531250\) | \(182400000000000000\) | \([2]\) | \(1179648\) | \(2.2074\) | |
91200.hi2 | 91200hk2 | \([0, 1, 0, -81633, 1864863]\) | \(14688124849/8122500\) | \(33269760000000000\) | \([2, 2]\) | \(589824\) | \(1.8608\) | |
91200.hi3 | 91200hk1 | \([0, 1, 0, -49633, -4247137]\) | \(3301293169/22800\) | \(93388800000000\) | \([2]\) | \(294912\) | \(1.5142\) | \(\Gamma_0(N)\)-optimal |
91200.hi4 | 91200hk3 | \([0, 1, 0, 318367, 15064863]\) | \(871257511151/527800050\) | \(-2161869004800000000\) | \([2]\) | \(1179648\) | \(2.2074\) |
Rank
sage: E.rank()
The elliptic curves in class 91200.hi have rank \(1\).
Complex multiplication
The elliptic curves in class 91200.hi do not have complex multiplication.Modular form 91200.2.a.hi
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.