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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 485520p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485520.p4 | 485520p1 | \([0, -1, 0, -4431, -32550]\) | \(24918016/13125\) | \(5068889490000\) | \([2]\) | \(655360\) | \(1.1295\) | \(\Gamma_0(N)\)-optimal* |
485520.p2 | 485520p2 | \([0, -1, 0, -40556, 3132000]\) | \(1193895376/11025\) | \(68125874745600\) | \([2, 2]\) | \(1310720\) | \(1.4760\) | \(\Gamma_0(N)\)-optimal* |
485520.p1 | 485520p3 | \([0, -1, 0, -647456, 200738640]\) | \(1214399773444/105\) | \(2595271418880\) | \([2]\) | \(2621440\) | \(1.8226\) | \(\Gamma_0(N)\)-optimal* |
485520.p3 | 485520p4 | \([0, -1, 0, -11656, 7478560]\) | \(-7086244/972405\) | \(-24034808610247680\) | \([2]\) | \(2621440\) | \(1.8226\) |
Rank
sage: E.rank()
The elliptic curves in class 485520p have rank \(1\).
Complex multiplication
The elliptic curves in class 485520p do not have complex multiplication.Modular form 485520.2.a.p
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 Cremona 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 Cremona labels.