Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 3432d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3432.e4 | 3432d1 | \([0, 1, 0, 36, 0]\) | \(19600688/11583\) | \(-2965248\) | \([2]\) | \(512\) | \(-0.068769\) | \(\Gamma_0(N)\)-optimal |
| 3432.e3 | 3432d2 | \([0, 1, 0, -144, -144]\) | \(324730948/184041\) | \(188457984\) | \([2, 2]\) | \(1024\) | \(0.27780\) | |
| 3432.e1 | 3432d3 | \([0, 1, 0, -1704, -27600]\) | \(267335955794/570999\) | \(1169405952\) | \([2]\) | \(2048\) | \(0.62438\) | |
| 3432.e2 | 3432d4 | \([0, 1, 0, -1464, 20976]\) | \(169556172914/942513\) | \(1930266624\) | \([2]\) | \(2048\) | \(0.62438\) |
Rank
sage: E.rank()
The elliptic curves in class 3432d have rank \(1\).
Complex multiplication
The elliptic curves in class 3432d do not have complex multiplication.Modular form 3432.2.a.d
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.
