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SageMath
E = EllipticCurve("gc1")
E.isogeny_class()
Elliptic curves in class 265200gc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
265200.gc2 | 265200gc1 | \([0, 1, 0, 1117, -166512]\) | \(615962624/48481875\) | \(-12120468750000\) | \([2]\) | \(589824\) | \(1.1900\) | \(\Gamma_0(N)\)-optimal |
265200.gc1 | 265200gc2 | \([0, 1, 0, -39508, -2929012]\) | \(1705021456336/68471325\) | \(273885300000000\) | \([2]\) | \(1179648\) | \(1.5366\) |
Rank
sage: E.rank()
The elliptic curves in class 265200gc have rank \(1\).
Complex multiplication
The elliptic curves in class 265200gc do not have complex multiplication.Modular form 265200.2.a.gc
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.