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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 459186r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
459186.r3 | 459186r1 | \([1, 1, 0, -5904, -158928]\) | \(38272753/4368\) | \(2598188266128\) | \([2]\) | \(1161216\) | \(1.1145\) | \(\Gamma_0(N)\)-optimal* |
459186.r2 | 459186r2 | \([1, 1, 0, -22724, 1142940]\) | \(2181825073/298116\) | \(177326349163236\) | \([2, 2]\) | \(2322432\) | \(1.4610\) | \(\Gamma_0(N)\)-optimal* |
459186.r1 | 459186r3 | \([1, 1, 0, -350714, 79794942]\) | \(8020417344913/187278\) | \(111397321910238\) | \([2]\) | \(4644864\) | \(1.8076\) | \(\Gamma_0(N)\)-optimal* |
459186.r4 | 459186r4 | \([1, 1, 0, 36146, 6146890]\) | \(8780064047/32388174\) | \(-19265241219805854\) | \([2]\) | \(4644864\) | \(1.8076\) |
Rank
sage: E.rank()
The elliptic curves in class 459186r have rank \(0\).
Complex multiplication
The elliptic curves in class 459186r do not have complex multiplication.Modular form 459186.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.