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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 3675.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
3675.i1 | 3675i2 | \([0, 1, 1, -5133, 140144]\) | \(-19539165184/46875\) | \(-35888671875\) | \([]\) | \(3456\) | \(0.90464\) | |
3675.i2 | 3675i1 | \([0, 1, 1, 117, 1019]\) | \(229376/675\) | \(-516796875\) | \([]\) | \(1152\) | \(0.35533\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 3675.i have rank \(1\).
Complex multiplication
The elliptic curves in class 3675.i do not have complex multiplication.Modular form 3675.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.