Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 207214.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
207214.k1 | 207214x2 | \([1, 0, 1, -8134129623, 282366846299844]\) | \(1265130637332599028485234161/3519232031977774\) | \(165565371387814550248894\) | \([]\) | \(144000000\) | \(4.1131\) | |
207214.k2 | 207214x1 | \([1, 0, 1, -28954013, -33860507816]\) | \(57059554959491530321/22493998606231264\) | \(1058249981642921904623584\) | \([]\) | \(28800000\) | \(3.3084\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 207214.k have rank \(1\).
Complex multiplication
The elliptic curves in class 207214.k do not have complex multiplication.Modular form 207214.2.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.