Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 53312bw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53312.p2 | 53312bw1 | \([0, 1, 0, -212, 2470]\) | \(-140608/289\) | \(-2176035904\) | \([2]\) | \(23040\) | \(0.48094\) | \(\Gamma_0(N)\)-optimal |
53312.p1 | 53312bw2 | \([0, 1, 0, -4377, 109927]\) | \(19248832/17\) | \(8192135168\) | \([2]\) | \(46080\) | \(0.82751\) |
Rank
sage: E.rank()
The elliptic curves in class 53312bw have rank \(1\).
Complex multiplication
The elliptic curves in class 53312bw do not have complex multiplication.Modular form 53312.2.a.bw
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.