Show commands:
SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 8470.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8470.e1 | 8470l4 | \([1, 0, 1, -425923, 106370856]\) | \(4823468134087681/30382271150\) | \(53824046660765150\) | \([2]\) | \(138240\) | \(2.0478\) | |
8470.e2 | 8470l2 | \([1, 0, 1, -32673, -2178244]\) | \(2177286259681/105875000\) | \(187564020875000\) | \([2]\) | \(46080\) | \(1.4985\) | |
8470.e3 | 8470l3 | \([1, 0, 1, -10893, 3609428]\) | \(-80677568161/3131816380\) | \(-5548203757969180\) | \([2]\) | \(69120\) | \(1.7012\) | |
8470.e4 | 8470l1 | \([1, 0, 1, 1207, -131892]\) | \(109902239/4312000\) | \(-7638971032000\) | \([2]\) | \(23040\) | \(1.1519\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8470.e have rank \(1\).
Complex multiplication
The elliptic curves in class 8470.e do not have complex multiplication.Modular form 8470.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.