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SageMath
E = EllipticCurve("lc1")
E.isogeny_class()
Elliptic curves in class 458640lc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
458640.lc4 | 458640lc1 | \([0, 0, 0, 76293, -6331094]\) | \(3774555693/3515200\) | \(-45736401508761600\) | \([2]\) | \(3981312\) | \(1.8840\) | \(\Gamma_0(N)\)-optimal* |
458640.lc3 | 458640lc2 | \([0, 0, 0, -394107, -57040214]\) | \(520300455507/193072360\) | \(2512071852868730880\) | \([2]\) | \(7962624\) | \(2.2306\) | \(\Gamma_0(N)\)-optimal* |
458640.lc2 | 458640lc3 | \([0, 0, 0, -1758267, -907263126]\) | \(-63378025803/812500\) | \(-7706600568576000000\) | \([2]\) | \(11943936\) | \(2.4333\) | |
458640.lc1 | 458640lc4 | \([0, 0, 0, -28218267, -57695715126]\) | \(261984288445803/42250\) | \(400743229565952000\) | \([2]\) | \(23887872\) | \(2.7799\) |
Rank
sage: E.rank()
The elliptic curves in class 458640lc have rank \(0\).
Complex multiplication
The elliptic curves in class 458640lc do not have complex multiplication.Modular form 458640.2.a.lc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.