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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 81600gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 81600.dz2 | 81600gk1 | \([0, -1, 0, -408833, -100442463]\) | \(1845026709625/793152\) | \(3248750592000000\) | \([2]\) | \(663552\) | \(1.9366\) | \(\Gamma_0(N)\)-optimal |
| 81600.dz3 | 81600gk2 | \([0, -1, 0, -344833, -133018463]\) | \(-1107111813625/1228691592\) | \(-5032720760832000000\) | \([2]\) | \(1327104\) | \(2.2831\) | |
| 81600.dz1 | 81600gk3 | \([0, -1, 0, -1200833, 383325537]\) | \(46753267515625/11591221248\) | \(47477642231808000000\) | \([2]\) | \(1990656\) | \(2.4859\) | |
| 81600.dz4 | 81600gk4 | \([0, -1, 0, 2895167, 2427229537]\) | \(655215969476375/1001033261568\) | \(-4100232239382528000000\) | \([2]\) | \(3981312\) | \(2.8324\) |
Rank
sage: E.rank()
The elliptic curves in class 81600gk have rank \(1\).
Complex multiplication
The elliptic curves in class 81600gk do not have complex multiplication.Modular form 81600.2.a.gk
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.
