Show commands:
SageMath
E = EllipticCurve("ga1")
E.isogeny_class()
Elliptic curves in class 137280ga
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
137280.cv4 | 137280ga1 | \([0, -1, 0, 70015, -15419775]\) | \(144794100308831/474439680000\) | \(-124371515473920000\) | \([2]\) | \(1179648\) | \(1.9633\) | \(\Gamma_0(N)\)-optimal |
137280.cv3 | 137280ga2 | \([0, -1, 0, -667265, -181012863]\) | \(125337052492018849/18404100000000\) | \(4824524390400000000\) | \([2, 2]\) | \(2359296\) | \(2.3099\) | |
137280.cv2 | 137280ga3 | \([0, -1, 0, -2863745, 1687313025]\) | \(9908022260084596129/1047363281250000\) | \(274560000000000000000\) | \([2]\) | \(4718592\) | \(2.6565\) | |
137280.cv1 | 137280ga4 | \([0, -1, 0, -10267265, -12659092863]\) | \(456612868287073618849/12544848030000\) | \(3288556641976320000\) | \([2]\) | \(4718592\) | \(2.6565\) |
Rank
sage: E.rank()
The elliptic curves in class 137280ga have rank \(1\).
Complex multiplication
The elliptic curves in class 137280ga do not have complex multiplication.Modular form 137280.2.a.ga
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.