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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 13680r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13680.g2 | 13680r1 | \([0, 0, 0, -234318, -43693117]\) | \(-121981271658244096/115966796875\) | \(-1352636718750000\) | \([2]\) | \(107520\) | \(1.8253\) | \(\Gamma_0(N)\)-optimal |
| 13680.g1 | 13680r2 | \([0, 0, 0, -3749943, -2795021242]\) | \(31248575021659890256/28203125\) | \(5263380000000\) | \([2]\) | \(215040\) | \(2.1719\) |
Rank
sage: E.rank()
The elliptic curves in class 13680r have rank \(1\).
Complex multiplication
The elliptic curves in class 13680r do not have complex multiplication.Modular form 13680.2.a.r
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.
