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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 140679.z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 140679.z1 | 140679y3 | \([1, -1, 0, -23529858, 43932670109]\) | \(16798320881842096017/2132227789307\) | \(182872906577266668147\) | \([2]\) | \(8257536\) | \(2.9098\) | |
| 140679.z2 | 140679y4 | \([1, -1, 0, -9334068, -10525526425]\) | \(1048626554636928177/48569076788309\) | \(4165581316684401079389\) | \([2]\) | \(8257536\) | \(2.9098\) | |
| 140679.z3 | 140679y2 | \([1, -1, 0, -1596723, 562088960]\) | \(5249244962308257/1448621666569\) | \(124242661138178508849\) | \([2, 2]\) | \(4128768\) | \(2.5632\) | |
| 140679.z4 | 140679y1 | \([1, -1, 0, 257682, 57319919]\) | \(22062729659823/29354283343\) | \(-2517603017064022503\) | \([2]\) | \(2064384\) | \(2.2166\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 140679.z have rank \(0\).
Complex multiplication
The elliptic curves in class 140679.z do not have complex multiplication.Modular form 140679.2.a.z
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
