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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 75150.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75150.m1 | 75150r1 | \([1, -1, 0, -10242, 366916]\) | \(83453453/8016\) | \(11413406250000\) | \([2]\) | \(230400\) | \(1.2431\) | \(\Gamma_0(N)\)-optimal |
75150.m2 | 75150r2 | \([1, -1, 0, 12258, 1739416]\) | \(143055667/1004004\) | \(-1429529132812500\) | \([2]\) | \(460800\) | \(1.5896\) |
Rank
sage: E.rank()
The elliptic curves in class 75150.m have rank \(1\).
Complex multiplication
The elliptic curves in class 75150.m do not have complex multiplication.Modular form 75150.2.a.m
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.