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Properties

Label 257754.bb
Number of curves $6$
Conductor $257754$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bb1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 257754.bb have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(7\)\(1 - T\)
\(17\)\(1 - T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 257754.bb do not have complex multiplication.

Modular form 257754.2.a.bb

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + 2 q^{10} + 4 q^{11} + q^{12} + 2 q^{13} - q^{14} - 2 q^{15} + q^{16} + q^{17} - q^{18} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 257754.bb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257754.bb1 257754bb5 \([1, 0, 1, -4952416052, 134144366143754]\) \(285531136548675601769470657/17941034271597192\) \(844051763358483174766152\) \([2]\) \(212336640\) \(4.0513\)  
257754.bb2 257754bb3 \([1, 0, 1, -310114892, 2087610425930]\) \(70108386184777836280897/552468975892674624\) \(25991389696038639132423744\) \([2, 2]\) \(106168320\) \(3.7048\)  
257754.bb3 257754bb6 \([1, 0, 1, -105630052, 4799570167946]\) \(-2770540998624539614657/209924951154647363208\) \(-9876104270952352446447346248\) \([2]\) \(212336640\) \(4.0513\)  
257754.bb4 257754bb2 \([1, 0, 1, -32751372, -18133417910]\) \(82582985847542515777/44772582831427584\) \(2106365603949985176981504\) \([2, 2]\) \(53084160\) \(3.3582\)  
257754.bb5 257754bb1 \([1, 0, 1, -25358092, -49090559926]\) \(38331145780597164097/55468445663232\) \(2609561893927378747392\) \([2]\) \(26542080\) \(3.0116\) \(\Gamma_0(N)\)-optimal
257754.bb6 257754bb4 \([1, 0, 1, 126319668, -142590599606]\) \(4738217997934888496063/2928751705237796928\) \(-137785704203164470976853568\) \([2]\) \(106168320\) \(3.7048\)