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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 400710.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400710.d1 | 400710d2 | \([1, 1, 0, -2658417003, 52756219896957]\) | \(-44164307457093068844199489/1823508000000000\) | \(-85788540370548000000000\) | \([]\) | \(169361280\) | \(3.8870\) | \(\Gamma_0(N)\)-optimal* |
400710.d2 | 400710d1 | \([1, 1, 0, -30105963, 84820077693]\) | \(-64144540676215729729/28962038218752000\) | \(-1362544603556858560512000\) | \([]\) | \(56453760\) | \(3.3376\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 400710.d have rank \(0\).
Complex multiplication
The elliptic curves in class 400710.d do not have complex multiplication.Modular form 400710.2.a.d
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.