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Properties

Label 86394.d
Number of curves $6$
Conductor $86394$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 86394.d have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(7\)\(1 + T\)
\(11\)\(1\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 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 86394.d do not have complex multiplication.

Modular form 86394.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + 2 q^{10} - q^{12} + 2 q^{13} + q^{14} + 2 q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 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 86394.d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86394.d1 86394b6 \([1, 1, 0, -1659951086, 26030318798796]\) \(285531136548675601769470657/17941034271597192\) \(31783636615224993056712\) \([2]\) \(39321600\) \(3.7781\)  
86394.d2 86394b4 \([1, 1, 0, -103944326, 405065871060]\) \(70108386184777836280897/552468975892674624\) \(978732491401402549568064\) \([2, 2]\) \(19660800\) \(3.4315\)  
86394.d3 86394b5 \([1, 1, 0, -35405086, 931351279324]\) \(-2770540998624539614657/209924951154647363208\) \(-371894856392478237412127688\) \([2]\) \(39321600\) \(3.7781\)  
86394.d4 86394b2 \([1, 1, 0, -10977606, -3522863340]\) \(82582985847542515777/44772582831427584\) \(79317361613426682138624\) \([2, 2]\) \(9830400\) \(3.0849\)  
86394.d5 86394b1 \([1, 1, 0, -8499526, -9529233644]\) \(38331145780597164097/55468445663232\) \(98265735067600945152\) \([2]\) \(4915200\) \(2.7384\) \(\Gamma_0(N)\)-optimal
86394.d6 86394b3 \([1, 1, 0, 42339834, -27654336684]\) \(4738217997934888496063/2928751705237796928\) \(-5188462299682776763564608\) \([2]\) \(19660800\) \(3.4315\)