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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 129150cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129150.cl2 | 129150cr1 | \([1, -1, 1, -4354355, -3218239853]\) | \(801581275315909089/70810888830976\) | \(806580280590336000000\) | \([]\) | \(6914880\) | \(2.7518\) | \(\Gamma_0(N)\)-optimal |
129150.cl1 | 129150cr2 | \([1, -1, 1, -2162545355, 38708114434147]\) | \(98191033604529537629349729/10906239337336\) | \(124228882451842875000\) | \([]\) | \(48404160\) | \(3.7248\) |
Rank
sage: E.rank()
The elliptic curves in class 129150cr have rank \(0\).
Complex multiplication
The elliptic curves in class 129150cr do not have complex multiplication.Modular form 129150.2.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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.