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SageMath
E = EllipticCurve("dn1")
E.isogeny_class()
Elliptic curves in class 493680dn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.dn3 | 493680dn1 | \([0, -1, 0, -84985600, 510400970752]\) | \(-9354997870579612441/10093752054144000\) | \(-73243432889513569419264000\) | \([2]\) | \(132710400\) | \(3.6579\) | \(\Gamma_0(N)\)-optimal* |
493680.dn2 | 493680dn2 | \([0, -1, 0, -1606836480, 24783313766400]\) | \(63229930193881628103961/26218934428500000\) | \(190252817183079991296000000\) | \([2]\) | \(265420800\) | \(4.0044\) | \(\Gamma_0(N)\)-optimal* |
493680.dn4 | 493680dn3 | \([0, -1, 0, 712307600, -9217952449088]\) | \(5508208700580085578359/8246033269590589440\) | \(-59835805471167177600679280640\) | \([2]\) | \(398131200\) | \(4.2072\) | |
493680.dn1 | 493680dn4 | \([0, -1, 0, -4679994480, -92617453339200]\) | \(1562225332123379392365961/393363080510106009600\) | \(2854365971544325785489349017600\) | \([2]\) | \(796262400\) | \(4.5538\) |
Rank
sage: E.rank()
The elliptic curves in class 493680dn have rank \(0\).
Complex multiplication
The elliptic curves in class 493680dn do not have complex multiplication.Modular form 493680.2.a.dn
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.