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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 58443a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
58443.o2 | 58443a1 | \([1, 1, 0, -10045, -412496]\) | \(-63282696625/4032567\) | \(-7143938427087\) | \([2]\) | \(130560\) | \(1.2199\) | \(\Gamma_0(N)\)-optimal |
58443.o1 | 58443a2 | \([1, 1, 0, -163110, -25423317]\) | \(270902819202625/1004157\) | \(1778925379077\) | \([2]\) | \(261120\) | \(1.5664\) |
Rank
sage: E.rank()
The elliptic curves in class 58443a have rank \(2\).
Complex multiplication
The elliptic curves in class 58443a do not have complex multiplication.Modular form 58443.2.a.a
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.