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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 219351i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
219351.i1 | 219351i1 | \([1, 1, 0, -944313, -353591280]\) | \(3858147330331321/45405657\) | \(1095982178827833\) | \([2]\) | \(3981312\) | \(2.0361\) | \(\Gamma_0(N)\)-optimal |
219351.i2 | 219351i2 | \([1, 1, 0, -919748, -372825675]\) | \(-3564818951887081/419636411073\) | \(-10129002827186901537\) | \([2]\) | \(7962624\) | \(2.3827\) |
Rank
sage: E.rank()
The elliptic curves in class 219351i have rank \(1\).
Complex multiplication
The elliptic curves in class 219351i do not have complex multiplication.Modular form 219351.2.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 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.