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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 33600gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.gh4 | 33600gk1 | \([0, 1, 0, 867, 81363]\) | \(4499456/180075\) | \(-2881200000000\) | \([2]\) | \(49152\) | \(1.0706\) | \(\Gamma_0(N)\)-optimal |
| 33600.gh3 | 33600gk2 | \([0, 1, 0, -23633, 1330863]\) | \(5702413264/275625\) | \(70560000000000\) | \([2, 2]\) | \(98304\) | \(1.4172\) | |
| 33600.gh2 | 33600gk3 | \([0, 1, 0, -65633, -4759137]\) | \(30534944836/8203125\) | \(8400000000000000\) | \([2]\) | \(196608\) | \(1.7638\) | |
| 33600.gh1 | 33600gk4 | \([0, 1, 0, -373633, 87780863]\) | \(5633270409316/14175\) | \(14515200000000\) | \([2]\) | \(196608\) | \(1.7638\) |
Rank
sage: E.rank()
The elliptic curves in class 33600gk have rank \(0\).
Complex multiplication
The elliptic curves in class 33600gk do not have complex multiplication.Modular form 33600.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
