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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 418950.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
418950.x1 | 418950x3 | \([1, -1, 0, -19560318192, 1052963399880216]\) | \(617611911727813844500009/1197723879765000\) | \(1605064549945538149453125000\) | \([2]\) | \(594542592\) | \(4.4737\) | \(\Gamma_0(N)\)-optimal* |
418950.x2 | 418950x4 | \([1, -1, 0, -3288300192, -51232163093784]\) | \(2934284984699764805929/851931751022747640\) | \(1141670025655606997573506875000\) | \([2]\) | \(594542592\) | \(4.4737\) | |
418950.x3 | 418950x2 | \([1, -1, 0, -1235445192, 16087110921216]\) | \(155617476551393929129/6633105589454400\) | \(8888995884233162412225000000\) | \([2, 2]\) | \(297271296\) | \(4.1271\) | \(\Gamma_0(N)\)-optimal* |
418950.x4 | 418950x1 | \([1, -1, 0, 38162808, 937543761216]\) | \(4586790226340951/286015269335040\) | \(-383287815588072346560000000\) | \([2]\) | \(148635648\) | \(3.7805\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 418950.x have rank \(1\).
Complex multiplication
The elliptic curves in class 418950.x do not have complex multiplication.Modular form 418950.2.a.x
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.