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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 416025.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
416025.k1 | 416025k3 | \([1, -1, 1, -286441880, 1866030400122]\) | \(36097320816649/80625\) | \(5805344750888759765625\) | \([2]\) | \(62447616\) | \(3.4213\) | \(\Gamma_0(N)\)-optimal* |
416025.k2 | 416025k4 | \([1, -1, 1, -49307630, -95968461378]\) | \(184122897769/51282015\) | \(3692524360871300981484375\) | \([2]\) | \(62447616\) | \(3.4213\) | |
416025.k3 | 416025k2 | \([1, -1, 1, -18105755, 28464616122]\) | \(9116230969/416025\) | \(29955578914586000390625\) | \([2, 2]\) | \(31223808\) | \(3.0747\) | \(\Gamma_0(N)\)-optimal* |
416025.k4 | 416025k1 | \([1, -1, 1, 615370, 1693407372]\) | \(357911/17415\) | \(-1253954466191972109375\) | \([2]\) | \(15611904\) | \(2.7281\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 416025.k have rank \(1\).
Complex multiplication
The elliptic curves in class 416025.k do not have complex multiplication.Modular form 416025.2.a.k
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.