Show commands:
SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 56784bl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
56784.bd3 | 56784bl1 | \([0, -1, 0, -152832, 22413312]\) | \(19968681097/628992\) | \(12435555313778688\) | \([2]\) | \(387072\) | \(1.8631\) | \(\Gamma_0(N)\)-optimal |
56784.bd2 | 56784bl2 | \([0, -1, 0, -369152, -54942720]\) | \(281397674377/96589584\) | \(1909634962872139776\) | \([2, 2]\) | \(774144\) | \(2.2097\) | |
56784.bd4 | 56784bl3 | \([0, -1, 0, 1091008, -383186688]\) | \(7264187703863/7406095788\) | \(-146423028958742495232\) | \([2]\) | \(1548288\) | \(2.5563\) | |
56784.bd1 | 56784bl4 | \([0, -1, 0, -5290432, -4680945920]\) | \(828279937799497/193444524\) | \(3824516175642279936\) | \([2]\) | \(1548288\) | \(2.5563\) |
Rank
sage: E.rank()
The elliptic curves in class 56784bl have rank \(0\).
Complex multiplication
The elliptic curves in class 56784bl do not have complex multiplication.Modular form 56784.2.a.bl
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.