Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 1575.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1575.k1 | 1575k2 | \([0, 0, 1, -1335, -18869]\) | \(-2887553024/16807\) | \(-1531537875\) | \([]\) | \(1200\) | \(0.60350\) | |
1575.k2 | 1575k1 | \([0, 0, 1, 15, 31]\) | \(4096/7\) | \(-637875\) | \([]\) | \(240\) | \(-0.20122\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1575.k have rank \(0\).
Complex multiplication
The elliptic curves in class 1575.k do not have complex multiplication.Modular form 1575.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.