Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 15680f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15680.bv4 | 15680f1 | \([0, 0, 0, 7252, 378672]\) | \(1367631/2800\) | \(-86354742476800\) | \([2]\) | \(36864\) | \(1.3585\) | \(\Gamma_0(N)\)-optimal |
| 15680.bv3 | 15680f2 | \([0, 0, 0, -55468, 4066608]\) | \(611960049/122500\) | \(3778019983360000\) | \([2, 2]\) | \(73728\) | \(1.7051\) | |
| 15680.bv2 | 15680f3 | \([0, 0, 0, -274988, -51867088]\) | \(74565301329/5468750\) | \(168661606400000000\) | \([2]\) | \(147456\) | \(2.0517\) | |
| 15680.bv1 | 15680f4 | \([0, 0, 0, -839468, 296028208]\) | \(2121328796049/120050\) | \(3702459583692800\) | \([2]\) | \(147456\) | \(2.0517\) |
Rank
sage: E.rank()
The elliptic curves in class 15680f have rank \(2\).
Complex multiplication
The elliptic curves in class 15680f do not have complex multiplication.Modular form 15680.2.a.f
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.
