Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 25270k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25270.a1 | 25270k1 | \([1, 0, 1, -7038, -170512]\) | \(5619814620139/1433600000\) | \(9833062400000\) | \([2]\) | \(57600\) | \(1.2026\) | \(\Gamma_0(N)\)-optimal |
25270.a2 | 25270k2 | \([1, 0, 1, 17282, -1084944]\) | \(83230218613781/122500000000\) | \(-840227500000000\) | \([2]\) | \(115200\) | \(1.5492\) |
Rank
sage: E.rank()
The elliptic curves in class 25270k have rank \(1\).
Complex multiplication
The elliptic curves in class 25270k do not have complex multiplication.Modular form 25270.2.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 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.