Show commands:
SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 129285n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
129285.g2 | 129285n1 | \([1, -1, 1, -25382, 1661716]\) | \(-19034163/1445\) | \(-137283787835415\) | \([2]\) | \(442368\) | \(1.4609\) | \(\Gamma_0(N)\)-optimal |
129285.g1 | 129285n2 | \([1, -1, 1, -413237, 102348874]\) | \(82142689923/425\) | \(40377584657475\) | \([2]\) | \(884736\) | \(1.8075\) |
Rank
sage: E.rank()
The elliptic curves in class 129285n have rank \(1\).
Complex multiplication
The elliptic curves in class 129285n do not have complex multiplication.Modular form 129285.2.a.n
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.