Show commands:
SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 327600gd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 327600.gd4 | 327600gd1 | \([0, 0, 0, 9825, -4473250]\) | \(35969456/2985255\) | \(-8705003580000000\) | \([2]\) | \(1966080\) | \(1.7381\) | \(\Gamma_0(N)\)-optimal |
| 327600.gd3 | 327600gd2 | \([0, 0, 0, -354675, -78466750]\) | \(423026849956/16769025\) | \(195593907600000000\) | \([2, 2]\) | \(3932160\) | \(2.0846\) | |
| 327600.gd2 | 327600gd3 | \([0, 0, 0, -921675, 235084250]\) | \(3711757787138/1124589375\) | \(26234420940000000000\) | \([2]\) | \(7864320\) | \(2.4312\) | |
| 327600.gd1 | 327600gd4 | \([0, 0, 0, -5619675, -5127601750]\) | \(841356017734178/1404585\) | \(32766158880000000\) | \([2]\) | \(7864320\) | \(2.4312\) |
Rank
sage: E.rank()
The elliptic curves in class 327600gd have rank \(1\).
Complex multiplication
The elliptic curves in class 327600gd do not have complex multiplication.Modular form 327600.2.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 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.
