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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 463680.dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.dk1 | 463680dk4 | \([0, 0, 0, -2909388, -1910045072]\) | \(14251520160844849/264449745\) | \(50537133271941120\) | \([2]\) | \(7864320\) | \(2.3301\) | |
463680.dk2 | 463680dk2 | \([0, 0, 0, -187788, -27786512]\) | \(3832302404449/472410225\) | \(90278999890329600\) | \([2, 2]\) | \(3932160\) | \(1.9835\) | |
463680.dk3 | 463680dk1 | \([0, 0, 0, -46668, 3429232]\) | \(58818484369/7455105\) | \(1424692751892480\) | \([2]\) | \(1966080\) | \(1.6369\) | \(\Gamma_0(N)\)-optimal* |
463680.dk4 | 463680dk3 | \([0, 0, 0, 275892, -143335568]\) | \(12152722588271/53476250625\) | \(-10219470639759360000\) | \([2]\) | \(7864320\) | \(2.3301\) |
Rank
sage: E.rank()
The elliptic curves in class 463680.dk have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.dk do not have complex multiplication.Modular form 463680.2.a.dk
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.