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SageMath
E = EllipticCurve("bb1")
E.isogeny_class()
Elliptic curves in class 257754bb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 257754.bb5 | 257754bb1 | \([1, 0, 1, -25358092, -49090559926]\) | \(38331145780597164097/55468445663232\) | \(2609561893927378747392\) | \([2]\) | \(26542080\) | \(3.0116\) | \(\Gamma_0(N)\)-optimal |
| 257754.bb4 | 257754bb2 | \([1, 0, 1, -32751372, -18133417910]\) | \(82582985847542515777/44772582831427584\) | \(2106365603949985176981504\) | \([2, 2]\) | \(53084160\) | \(3.3582\) | |
| 257754.bb2 | 257754bb3 | \([1, 0, 1, -310114892, 2087610425930]\) | \(70108386184777836280897/552468975892674624\) | \(25991389696038639132423744\) | \([2, 2]\) | \(106168320\) | \(3.7048\) | |
| 257754.bb6 | 257754bb4 | \([1, 0, 1, 126319668, -142590599606]\) | \(4738217997934888496063/2928751705237796928\) | \(-137785704203164470976853568\) | \([2]\) | \(106168320\) | \(3.7048\) | |
| 257754.bb1 | 257754bb5 | \([1, 0, 1, -4952416052, 134144366143754]\) | \(285531136548675601769470657/17941034271597192\) | \(844051763358483174766152\) | \([2]\) | \(212336640\) | \(4.0513\) | |
| 257754.bb3 | 257754bb6 | \([1, 0, 1, -105630052, 4799570167946]\) | \(-2770540998624539614657/209924951154647363208\) | \(-9876104270952352446447346248\) | \([2]\) | \(212336640\) | \(4.0513\) |
Rank
sage: E.rank()
The elliptic curves in class 257754bb have rank \(1\).
Complex multiplication
The elliptic curves in class 257754bb do not have complex multiplication.Modular form 257754.2.a.bb
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
