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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 455175.ce
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
455175.ce1 | 455175ce4 | \([1, -1, 1, -109633505, -441791638378]\) | \(530044731605089/26309115\) | \(7233480614525095546875\) | \([2]\) | \(56623104\) | \(3.2658\) | |
455175.ce2 | 455175ce3 | \([1, -1, 1, -34854755, 73635526622]\) | \(17032120495489/1339001685\) | \(368147797113811635703125\) | \([2]\) | \(56623104\) | \(3.2658\) | \(\Gamma_0(N)\)-optimal* |
455175.ce3 | 455175ce2 | \([1, -1, 1, -7219130, -6120887128]\) | \(151334226289/28676025\) | \(7884243576385484765625\) | \([2, 2]\) | \(28311552\) | \(2.9192\) | \(\Gamma_0(N)\)-optimal* |
455175.ce4 | 455175ce1 | \([1, -1, 1, 908995, -561249628]\) | \(302111711/669375\) | \(-184039299168662109375\) | \([2]\) | \(14155776\) | \(2.5726\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 455175.ce have rank \(1\).
Complex multiplication
The elliptic curves in class 455175.ce do not have complex multiplication.Modular form 455175.2.a.ce
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.