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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 2175.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2175.c1 | 2175e2 | \([1, 1, 1, -243, 1356]\) | \(12698260037/7569\) | \(946125\) | \([2]\) | \(384\) | \(0.090849\) | |
2175.c2 | 2175e1 | \([1, 1, 1, -18, 6]\) | \(5177717/2349\) | \(293625\) | \([2]\) | \(192\) | \(-0.25572\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2175.c have rank \(1\).
Complex multiplication
The elliptic curves in class 2175.c do not have complex multiplication.Modular form 2175.2.a.c
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.