Show commands:
SageMath
E = EllipticCurve("hn1")
E.isogeny_class()
Elliptic curves in class 193600.hn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
193600.hn1 | 193600em2 | \([0, -1, 0, -83233, -9465663]\) | \(-128667913/4096\) | \(-2030043136000000\) | \([]\) | \(995328\) | \(1.7132\) | |
193600.hn2 | 193600em1 | \([0, -1, 0, 4767, -49663]\) | \(24167/16\) | \(-7929856000000\) | \([]\) | \(331776\) | \(1.1639\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 193600.hn have rank \(1\).
Complex multiplication
The elliptic curves in class 193600.hn do not have complex multiplication.Modular form 193600.2.a.hn
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.