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Properties

Label 2352c
Number of curves $6$
Conductor $2352$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("c1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 2352c have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 2352c do not have complex multiplication.

Modular form 2352.2.a.c

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 2352c

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2352.i5 2352c1 \([0, -1, 0, 33, 78]\) \(2048/3\) \(-5647152\) \([2]\) \(384\) \(-0.018971\) \(\Gamma_0(N)\)-optimal
2352.i4 2352c2 \([0, -1, 0, -212, 960]\) \(35152/9\) \(271063296\) \([2, 2]\) \(768\) \(0.32760\)  
2352.i3 2352c3 \([0, -1, 0, -1192, -14720]\) \(1556068/81\) \(9758278656\) \([2, 2]\) \(1536\) \(0.67418\)  
2352.i2 2352c4 \([0, -1, 0, -3152, 69168]\) \(28756228/3\) \(361417728\) \([2]\) \(1536\) \(0.67418\)  
2352.i1 2352c5 \([0, -1, 0, -18832, -988448]\) \(3065617154/9\) \(2168506368\) \([2]\) \(3072\) \(1.0208\)  
2352.i6 2352c6 \([0, -1, 0, 768, -60192]\) \(207646/6561\) \(-1580841142272\) \([2]\) \(3072\) \(1.0208\)