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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 72128i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72128.r1 | 72128i1 | \([0, 1, 0, -177, -785]\) | \(109744/23\) | \(129253376\) | \([2]\) | \(24576\) | \(0.27127\) | \(\Gamma_0(N)\)-optimal |
72128.r2 | 72128i2 | \([0, 1, 0, 383, -4257]\) | \(275684/529\) | \(-11891310592\) | \([2]\) | \(49152\) | \(0.61785\) |
Rank
sage: E.rank()
The elliptic curves in class 72128i have rank \(0\).
Complex multiplication
The elliptic curves in class 72128i do not have complex multiplication.Modular form 72128.2.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.