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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 2475.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2475.i1 | 2475j3 | \([1, -1, 0, -13317, -588034]\) | \(22930509321/6875\) | \(78310546875\) | \([2]\) | \(3072\) | \(1.0684\) | |
2475.i2 | 2475j4 | \([1, -1, 0, -6567, 201716]\) | \(2749884201/73205\) | \(833850703125\) | \([2]\) | \(3072\) | \(1.0684\) | |
2475.i3 | 2475j2 | \([1, -1, 0, -942, -6409]\) | \(8120601/3025\) | \(34456640625\) | \([2, 2]\) | \(1536\) | \(0.72179\) | |
2475.i4 | 2475j1 | \([1, -1, 0, 183, -784]\) | \(59319/55\) | \(-626484375\) | \([2]\) | \(768\) | \(0.37521\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2475.i have rank \(1\).
Complex multiplication
The elliptic curves in class 2475.i do not have complex multiplication.Modular form 2475.2.a.i
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.