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SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 81600.gd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81600.gd1 | 81600dw3 | \([0, 1, 0, -1200833, -383325537]\) | \(46753267515625/11591221248\) | \(47477642231808000000\) | \([2]\) | \(1990656\) | \(2.4859\) | |
| 81600.gd2 | 81600dw1 | \([0, 1, 0, -408833, 100442463]\) | \(1845026709625/793152\) | \(3248750592000000\) | \([2]\) | \(663552\) | \(1.9366\) | \(\Gamma_0(N)\)-optimal |
| 81600.gd3 | 81600dw2 | \([0, 1, 0, -344833, 133018463]\) | \(-1107111813625/1228691592\) | \(-5032720760832000000\) | \([2]\) | \(1327104\) | \(2.2831\) | |
| 81600.gd4 | 81600dw4 | \([0, 1, 0, 2895167, -2427229537]\) | \(655215969476375/1001033261568\) | \(-4100232239382528000000\) | \([2]\) | \(3981312\) | \(2.8324\) |
Rank
sage: E.rank()
The elliptic curves in class 81600.gd have rank \(1\).
Complex multiplication
The elliptic curves in class 81600.gd do not have complex multiplication.Modular form 81600.2.a.gd
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
