Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 3264x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3264.w2 | 3264x1 | \([0, 1, 0, -16353, -810081]\) | \(1845026709625/793152\) | \(207920037888\) | \([2]\) | \(4608\) | \(1.1318\) | \(\Gamma_0(N)\)-optimal |
| 3264.w3 | 3264x2 | \([0, 1, 0, -13793, -1069665]\) | \(-1107111813625/1228691592\) | \(-322094128693248\) | \([2]\) | \(9216\) | \(1.4784\) | |
| 3264.w1 | 3264x3 | \([0, 1, 0, -48033, 3047391]\) | \(46753267515625/11591221248\) | \(3038569102835712\) | \([2]\) | \(13824\) | \(1.6811\) | |
| 3264.w4 | 3264x4 | \([0, 1, 0, 115807, 19464159]\) | \(655215969476375/1001033261568\) | \(-262414863320481792\) | \([2]\) | \(27648\) | \(2.0277\) |
Rank
sage: E.rank()
The elliptic curves in class 3264x have rank \(1\).
Complex multiplication
The elliptic curves in class 3264x do not have complex multiplication.Modular form 3264.2.a.x
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.
