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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 4225.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4225.p1 | 4225c2 | \([0, -1, 1, -2147708, -1210749057]\) | \(23242854400/13\) | \(612778486328125\) | \([]\) | \(70560\) | \(2.1644\) | |
4225.p2 | 4225c1 | \([0, -1, 1, -16618, 764623]\) | \(4206161920/371293\) | \(44804009850925\) | \([]\) | \(14112\) | \(1.3597\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4225.p have rank \(1\).
Complex multiplication
The elliptic curves in class 4225.p do not have complex multiplication.Modular form 4225.2.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 LMFDB numbering.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.