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SageMath
E = EllipticCurve("t1")
E.isogeny_class()
Elliptic curves in class 259182t
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259182.t4 | 259182t1 | \([1, -1, 0, 65862, -2416716]\) | \(24464768327/16084992\) | \(-20773251949621248\) | \([2]\) | \(2580480\) | \(1.8199\) | \(\Gamma_0(N)\)-optimal |
259182.t3 | 259182t2 | \([1, -1, 0, -282618, -19771020]\) | \(1933038007993/986965056\) | \(1274633756346291264\) | \([2, 2]\) | \(5160960\) | \(2.1664\) | |
259182.t2 | 259182t3 | \([1, -1, 0, -2504178, 1511772444]\) | \(1344732056152633/14342440728\) | \(18522802797493041432\) | \([2]\) | \(10321920\) | \(2.5130\) | |
259182.t1 | 259182t4 | \([1, -1, 0, -3636738, -2666171700]\) | \(4118851966518073/4167363816\) | \(5382016883533609704\) | \([2]\) | \(10321920\) | \(2.5130\) |
Rank
sage: E.rank()
The elliptic curves in class 259182t have rank \(2\).
Complex multiplication
The elliptic curves in class 259182t do not have complex multiplication.Modular form 259182.2.a.t
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.