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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 471510.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
471510.d1 | 471510d3 | \([1, -1, 0, -353067480, 2553581378700]\) | \(1383277217333832812809/27202500\) | \(95718627158602500\) | \([2]\) | \(66060288\) | \(3.2435\) | \(\Gamma_0(N)\)-optimal* |
471510.d2 | 471510d2 | \([1, -1, 0, -22067460, 39901026816]\) | \(337748263783145929/47358464400\) | \(166642301138040608400\) | \([2, 2]\) | \(33030144\) | \(2.8969\) | \(\Gamma_0(N)\)-optimal* |
471510.d3 | 471510d4 | \([1, -1, 0, -20090160, 47340815796]\) | \(-254850956966062729/127607200177860\) | \(-449017039484522965331460\) | \([2]\) | \(66060288\) | \(3.2435\) | |
471510.d4 | 471510d1 | \([1, -1, 0, -1503540, 504668880]\) | \(106827039259849/30599112960\) | \(107670437820134242560\) | \([2]\) | \(16515072\) | \(2.5503\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 471510.d have rank \(1\).
Complex multiplication
The elliptic curves in class 471510.d do not have complex multiplication.Modular form 471510.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.