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SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 169050he
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.b1 | 169050he1 | \([1, 1, 0, -10667325, 11134072125]\) | \(584214157617173/104837677056\) | \(24089937242112000000000\) | \([2]\) | \(25804800\) | \(3.0138\) | \(\Gamma_0(N)\)-optimal |
169050.b2 | 169050he2 | \([1, 1, 0, 20692675, 64289272125]\) | \(4264374232864747/10236109169664\) | \(-2352085952542578000000000\) | \([2]\) | \(51609600\) | \(3.3604\) |
Rank
sage: E.rank()
The elliptic curves in class 169050he have rank \(1\).
Complex multiplication
The elliptic curves in class 169050he do not have complex multiplication.Modular form 169050.2.a.he
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.