Show commands:
SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 66248.p
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66248.p1 | 66248e4 | \([0, 0, 0, -2476019, -1499606290]\) | \(1443468546/7\) | \(8140973597259776\) | \([2]\) | \(884736\) | \(2.2534\) | |
66248.p2 | 66248e3 | \([0, 0, 0, -488579, 103992798]\) | \(11090466/2401\) | \(2792353943860103168\) | \([2]\) | \(884736\) | \(2.2534\) | |
66248.p3 | 66248e2 | \([0, 0, 0, -157339, -22607130]\) | \(740772/49\) | \(28493407590409216\) | \([2, 2]\) | \(442368\) | \(1.9068\) | |
66248.p4 | 66248e1 | \([0, 0, 0, 8281, -1507142]\) | \(432/7\) | \(-1017621699657472\) | \([2]\) | \(221184\) | \(1.5602\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 66248.p have rank \(0\).
Complex multiplication
The elliptic curves in class 66248.p do not have complex multiplication.Modular form 66248.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.