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Properties

Label 6936m
Number of curves $6$
Conductor $6936$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("m1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 6936m have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 + T + 7 T^{2}\) 1.7.b
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 5 T + 13 T^{2}\) 1.13.f
\(19\) \( 1 - 5 T + 19 T^{2}\) 1.19.af
\(23\) \( 1 + 2 T + 23 T^{2}\) 1.23.c
\(29\) \( 1 + 4 T + 29 T^{2}\) 1.29.e
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6936m do not have complex multiplication.

Modular form 6936.2.a.m

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + 2 q^{5} + q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 6936m

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6936.p5 6936m1 \([0, 1, 0, 193, 1338]\) \(2048/3\) \(-1158603312\) \([2]\) \(2560\) \(0.42468\) \(\Gamma_0(N)\)-optimal
6936.p4 6936m2 \([0, 1, 0, -1252, 12320]\) \(35152/9\) \(55612958976\) \([2, 2]\) \(5120\) \(0.77125\)  
6936.p3 6936m3 \([0, 1, 0, -7032, -218880]\) \(1556068/81\) \(2002066523136\) \([2, 2]\) \(10240\) \(1.1178\)  
6936.p2 6936m4 \([0, 1, 0, -18592, 969488]\) \(28756228/3\) \(74150611968\) \([2]\) \(10240\) \(1.1178\)  
6936.p1 6936m5 \([0, 1, 0, -111072, -14285088]\) \(3065617154/9\) \(444903671808\) \([2]\) \(20480\) \(1.4644\)  
6936.p6 6936m6 \([0, 1, 0, 4528, -856992]\) \(207646/6561\) \(-324334776748032\) \([2]\) \(20480\) \(1.4644\)