Show commands:
SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 78400bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78400.eh3 | 78400bp1 | \([0, -1, 0, -9473, -424703]\) | \(-121945/32\) | \(-24672783564800\) | \([]\) | \(138240\) | \(1.2866\) | \(\Gamma_0(N)\)-optimal |
78400.eh4 | 78400bp2 | \([0, -1, 0, 68927, 3134657]\) | \(46969655/32768\) | \(-25264930370355200\) | \([]\) | \(414720\) | \(1.8359\) | |
78400.eh2 | 78400bp3 | \([0, -1, 0, -40833, 37489537]\) | \(-25/2\) | \(-602362880000000000\) | \([]\) | \(691200\) | \(2.0913\) | |
78400.eh1 | 78400bp4 | \([0, -1, 0, -9840833, 11885689537]\) | \(-349938025/8\) | \(-2409451520000000000\) | \([]\) | \(2073600\) | \(2.6406\) |
Rank
sage: E.rank()
The elliptic curves in class 78400bp have rank \(0\).
Complex multiplication
The elliptic curves in class 78400bp do not have complex multiplication.Modular form 78400.2.a.bp
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.