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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 75150.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75150.x1 | 75150q2 | \([1, -1, 0, -4759542, -3835054634]\) | \(1046819248735488409/47650971093750\) | \(542774342614746093750\) | \([2]\) | \(4128768\) | \(2.7406\) | |
75150.x2 | 75150q1 | \([1, -1, 0, 161208, -228144884]\) | \(40675641638471/1996889557500\) | \(-22745820115898437500\) | \([2]\) | \(2064384\) | \(2.3940\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 75150.x have rank \(1\).
Complex multiplication
The elliptic curves in class 75150.x do not have complex multiplication.Modular form 75150.2.a.x
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.