Show commands:
SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 4290o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4290.p4 | 4290o1 | \([1, 0, 1, -598, -64744]\) | \(-23592983745241/1794399750000\) | \(-1794399750000\) | \([6]\) | \(6912\) | \(1.0310\) | \(\Gamma_0(N)\)-optimal |
4290.p3 | 4290o2 | \([1, 0, 1, -28098, -1802744]\) | \(2453170411237305241/19353090685500\) | \(19353090685500\) | \([6]\) | \(13824\) | \(1.3776\) | |
4290.p2 | 4290o3 | \([1, 0, 1, -141973, -20602144]\) | \(-316472948332146183241/7074906009600\) | \(-7074906009600\) | \([2]\) | \(20736\) | \(1.5803\) | |
4290.p1 | 4290o4 | \([1, 0, 1, -2271573, -1317954464]\) | \(1296294060988412126189641/647824320\) | \(647824320\) | \([2]\) | \(41472\) | \(1.9269\) |
Rank
sage: E.rank()
The elliptic curves in class 4290o have rank \(0\).
Complex multiplication
The elliptic curves in class 4290o do not have complex multiplication.Modular form 4290.2.a.o
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.