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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 242550cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
242550.cr2 | 242550cr1 | \([1, -1, 0, 766008, -33788084]\) | \(296740963/174636\) | \(-29253617787023437500\) | \([2]\) | \(5898240\) | \(2.4246\) | \(\Gamma_0(N)\)-optimal |
242550.cr1 | 242550cr2 | \([1, -1, 0, -3092742, -269171834]\) | \(19530306557/11114334\) | \(1861783817731277343750\) | \([2]\) | \(11796480\) | \(2.7712\) |
Rank
sage: E.rank()
The elliptic curves in class 242550cr have rank \(0\).
Complex multiplication
The elliptic curves in class 242550cr do not have complex multiplication.Modular form 242550.2.a.cr
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.