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Properties

Label 11760.p
Number of curves $8$
Conductor $11760$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("p1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 11760.p have rank \(1\).

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 \(\mathbb{F}_p\)
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 11760.2.a.p

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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 11760.p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
11760.p1 11760bq7 \([0, -1, 0, -1693456, -847656704]\) \(1114544804970241/405\) \(195165573120\) \([2]\) \(98304\) \(1.9570\)  
11760.p2 11760bq5 \([0, -1, 0, -105856, -13214144]\) \(272223782641/164025\) \(79042057113600\) \([2, 2]\) \(49152\) \(1.6104\)  
11760.p3 11760bq8 \([0, -1, 0, -86256, -18278784]\) \(-147281603041/215233605\) \(-103718987344465920\) \([2]\) \(98304\) \(1.9570\)  
11760.p4 11760bq4 \([0, -1, 0, -62736, 6069120]\) \(56667352321/15\) \(7228354560\) \([2]\) \(24576\) \(1.2638\)  
11760.p5 11760bq3 \([0, -1, 0, -7856, -121344]\) \(111284641/50625\) \(24395696640000\) \([2, 2]\) \(24576\) \(1.2638\)  
11760.p6 11760bq2 \([0, -1, 0, -3936, 95040]\) \(13997521/225\) \(108425318400\) \([2, 2]\) \(12288\) \(0.91725\)  
11760.p7 11760bq1 \([0, -1, 0, -16, 4096]\) \(-1/15\) \(-7228354560\) \([2]\) \(6144\) \(0.57068\) \(\Gamma_0(N)\)-optimal
11760.p8 11760bq6 \([0, -1, 0, 27424, -939840]\) \(4733169839/3515625\) \(-1694145600000000\) \([2]\) \(49152\) \(1.6104\)