Show commands:
SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 33810bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33810.be2 | 33810bd1 | \([1, 0, 1, -614, 10532]\) | \(-217081801/285660\) | \(-33607613340\) | \([2]\) | \(41472\) | \(0.71273\) | \(\Gamma_0(N)\)-optimal |
33810.be1 | 33810bd2 | \([1, 0, 1, -11884, 497396]\) | \(1577505447721/838350\) | \(98631039150\) | \([2]\) | \(82944\) | \(1.0593\) |
Rank
sage: E.rank()
The elliptic curves in class 33810bd have rank \(1\).
Complex multiplication
The elliptic curves in class 33810bd do not have complex multiplication.Modular form 33810.2.a.bd
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.