Show commands:
SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 6800.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6800.m1 | 6800a3 | \([0, 0, 0, -29075, 1867250]\) | \(84944038338/2088025\) | \(66816800000000\) | \([2]\) | \(12288\) | \(1.4366\) | |
| 6800.m2 | 6800a2 | \([0, 0, 0, -4075, -57750]\) | \(467720676/180625\) | \(2890000000000\) | \([2, 2]\) | \(6144\) | \(1.0900\) | |
| 6800.m3 | 6800a1 | \([0, 0, 0, -3575, -82250]\) | \(1263257424/425\) | \(1700000000\) | \([2]\) | \(3072\) | \(0.74344\) | \(\Gamma_0(N)\)-optimal |
| 6800.m4 | 6800a4 | \([0, 0, 0, 12925, -414750]\) | \(7462174302/6640625\) | \(-212500000000000\) | \([2]\) | \(12288\) | \(1.4366\) |
Rank
sage: E.rank()
The elliptic curves in class 6800.m have rank \(1\).
Complex multiplication
The elliptic curves in class 6800.m do not have complex multiplication.Modular form 6800.2.a.m
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 LMFDB 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 LMFDB labels.
