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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 17325.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
17325.i1 | 17325bk2 | \([1, -1, 1, -11750, 493152]\) | \(1968634623437/5929\) | \(540280125\) | \([2]\) | \(19968\) | \(0.90346\) | |
17325.i2 | 17325bk1 | \([1, -1, 1, -725, 8052]\) | \(-461889917/26411\) | \(-2406702375\) | \([2]\) | \(9984\) | \(0.55689\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 17325.i have rank \(1\).
Complex multiplication
The elliptic curves in class 17325.i do not have complex multiplication.Modular form 17325.2.a.i
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.