Show commands:
SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 64400.bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
64400.bj1 | 64400bj4 | \([0, 0, 0, -16835075, -26587074750]\) | \(8244966675515989329/3081640625\) | \(197225000000000000\) | \([2]\) | \(2211840\) | \(2.6698\) | |
64400.bj2 | 64400bj3 | \([0, 0, 0, -2207075, 643177250]\) | \(18577831198352049/7958740140575\) | \(509359368996800000000\) | \([2]\) | \(2211840\) | \(2.6698\) | |
64400.bj3 | 64400bj2 | \([0, 0, 0, -1057075, -411372750]\) | \(2041085246738049/38897700625\) | \(2489452840000000000\) | \([2, 2]\) | \(1105920\) | \(2.3232\) | |
64400.bj4 | 64400bj1 | \([0, 0, 0, 925, -18854750]\) | \(1367631/2399636575\) | \(-153576740800000000\) | \([2]\) | \(552960\) | \(1.9766\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 64400.bj have rank \(1\).
Complex multiplication
The elliptic curves in class 64400.bj do not have complex multiplication.Modular form 64400.2.a.bj
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.