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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 19992i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19992.e2 | 19992i1 | \([0, -1, 0, -464, -4836]\) | \(-31522396/12393\) | \(-4352818176\) | \([2]\) | \(13824\) | \(0.55858\) | \(\Gamma_0(N)\)-optimal |
19992.e1 | 19992i2 | \([0, -1, 0, -8024, -273972]\) | \(81344187038/7803\) | \(5481326592\) | \([2]\) | \(27648\) | \(0.90516\) |
Rank
sage: E.rank()
The elliptic curves in class 19992i have rank \(1\).
Complex multiplication
The elliptic curves in class 19992i do not have complex multiplication.Modular form 19992.2.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.