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Properties

Label 291312bp
Number of curves $6$
Conductor $291312$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 291312bp have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1 - T\)
\(17\)\(1\)
 
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.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 291312bp do not have complex multiplication.

Modular form 291312.2.a.bp

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{5} + q^{7} - 4 q^{11} - 2 q^{13} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.

Elliptic curves in class 291312bp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
291312.bp5 291312bp1 \([0, 0, 0, -2923275171, 60758772299426]\) \(38331145780597164097/55468445663232\) \(3997854653498821000531279872\) \([2]\) \(212336640\) \(4.1985\) \(\Gamma_0(N)\)-optimal
291312.bp4 291312bp2 \([0, 0, 0, -3775570851, 22441092658850]\) \(82582985847542515777/44772582831427584\) \(3226956812680852421648805003264\) \([2, 2]\) \(424673280\) \(4.5450\)  
291312.bp6 291312bp3 \([0, 0, 0, 14562103389, 176503229018786]\) \(4738217997934888496063/2928751705237796928\) \(-211088006770826419766401742143488\) \([2]\) \(849346560\) \(4.8916\)  
291312.bp2 291312bp4 \([0, 0, 0, -35749975971, -2583952540697950]\) \(70108386184777836280897/552468975892674624\) \(39818867101414332709082979631104\) \([2, 2]\) \(849346560\) \(4.8916\)  
291312.bp3 291312bp5 \([0, 0, 0, -12177008931, -5940634611545566]\) \(-2770540998624539614657/209924951154647363208\) \(-15130213814796461103815666178490368\) \([2]\) \(1698693120\) \(5.2382\)  
291312.bp1 291312bp6 \([0, 0, 0, -570913424931, -166036463004685534]\) \(285531136548675601769470657/17941034271597192\) \(1293089187801614706177862828032\) \([2]\) \(1698693120\) \(5.2382\)