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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 134310cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
134310.f2 | 134310cr1 | \([1, 1, 0, -138668, 31326288]\) | \(-166456688365729/143856000000\) | \(-254849679216000000\) | \([2]\) | \(1680000\) | \(2.0374\) | \(\Gamma_0(N)\)-optimal |
134310.f1 | 134310cr2 | \([1, 1, 0, -2558668, 1573834288]\) | \(1045706191321645729/323352324000\) | \(572838366457764000\) | \([2]\) | \(3360000\) | \(2.3839\) |
Rank
sage: E.rank()
The elliptic curves in class 134310cr have rank \(0\).
Complex multiplication
The elliptic curves in class 134310cr do not have complex multiplication.Modular form 134310.2.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.