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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 14157.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14157.r1 | 14157j3 | \([1, -1, 0, -75708, -7998809]\) | \(37159393753/1053\) | \(1359915771357\) | \([2]\) | \(40960\) | \(1.4299\) | |
14157.r2 | 14157j4 | \([1, -1, 0, -21258, 1085629]\) | \(822656953/85683\) | \(110656849987827\) | \([2]\) | \(40960\) | \(1.4299\) | |
14157.r3 | 14157j2 | \([1, -1, 0, -4923, -113360]\) | \(10218313/1521\) | \(1964322780849\) | \([2, 2]\) | \(20480\) | \(1.0833\) | |
14157.r4 | 14157j1 | \([1, -1, 0, 522, -9905]\) | \(12167/39\) | \(-50367250791\) | \([2]\) | \(10240\) | \(0.73674\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 14157.r have rank \(1\).
Complex multiplication
The elliptic curves in class 14157.r do not have complex multiplication.Modular form 14157.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.