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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 17238.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17238.e1 | 17238f5 | \([1, 0, 1, -4688740, -3908190232]\) | \(2361739090258884097/5202\) | \(25109060418\) | \([2]\) | \(294912\) | \(2.1296\) | |
| 17238.e2 | 17238f3 | \([1, 0, 1, -293050, -61082344]\) | \(576615941610337/27060804\) | \(130617332294436\) | \([2, 2]\) | \(147456\) | \(1.7830\) | |
| 17238.e3 | 17238f6 | \([1, 0, 1, -277840, -67701736]\) | \(-491411892194497/125563633938\) | \(-606071678364643842\) | \([2]\) | \(294912\) | \(2.1296\) | |
| 17238.e4 | 17238f2 | \([1, 0, 1, -19270, -850744]\) | \(163936758817/30338064\) | \(146436040357776\) | \([2, 2]\) | \(73728\) | \(1.4364\) | |
| 17238.e5 | 17238f1 | \([1, 0, 1, -5750, 155144]\) | \(4354703137/352512\) | \(1701508094208\) | \([2]\) | \(36864\) | \(1.0898\) | \(\Gamma_0(N)\)-optimal |
| 17238.e6 | 17238f4 | \([1, 0, 1, 38190, -4941896]\) | \(1276229915423/2927177028\) | \(-14128924423343652\) | \([2]\) | \(147456\) | \(1.7830\) |
Rank
sage: E.rank()
The elliptic curves in class 17238.e have rank \(1\).
Complex multiplication
The elliptic curves in class 17238.e do not have complex multiplication.Modular form 17238.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
