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Properties

Label 11760e
Number of curves $6$
Conductor $11760$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("e1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 11760e have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 3 T + 11 T^{2}\) 1.11.d
\(13\) \( 1 - 5 T + 13 T^{2}\) 1.13.af
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 5 T + 19 T^{2}\) 1.19.f
\(23\) \( 1 - 9 T + 23 T^{2}\) 1.23.aj
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 11760e do not have complex multiplication.

Modular form 11760.2.a.e

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} + 4 q^{11} + 2 q^{13} + q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 11760e

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
11760.o4 11760e1 \([0, -1, 0, -8591, 309366]\) \(37256083456/525\) \(988251600\) \([2]\) \(12288\) \(0.86615\) \(\Gamma_0(N)\)-optimal
11760.o3 11760e2 \([0, -1, 0, -8836, 291040]\) \(2533446736/275625\) \(8301313440000\) \([2, 2]\) \(24576\) \(1.2127\)  
11760.o2 11760e3 \([0, -1, 0, -33336, -2021760]\) \(34008619684/4862025\) \(585740676326400\) \([2, 2]\) \(49152\) \(1.5593\)  
11760.o5 11760e4 \([0, -1, 0, 11744, 1427056]\) \(1486779836/8203125\) \(-988251600000000\) \([2]\) \(49152\) \(1.5593\)  
11760.o1 11760e5 \([0, -1, 0, -513536, -141471840]\) \(62161150998242/1607445\) \(387306079856640\) \([2]\) \(98304\) \(1.9059\)  
11760.o6 11760e6 \([0, -1, 0, 54864, -10982880]\) \(75798394558/259416045\) \(-62505038393763840\) \([2]\) \(98304\) \(1.9059\)