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Properties

Label 360.b
Number of curves $6$
Conductor $360$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([0, 0, 0, -28803, 1881502]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 360.b have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 360.b do not have complex multiplication.

Modular form 360.2.a.b

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} + 4 q^{11} + 6 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 360.b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
360.b1 360a5 \([0, 0, 0, -28803, 1881502]\) \(1770025017602/75\) \(111974400\) \([2]\) \(512\) \(1.0293\)  
360.b2 360a3 \([0, 0, 0, -1803, 29302]\) \(868327204/5625\) \(4199040000\) \([2, 2]\) \(256\) \(0.68273\)  
360.b3 360a6 \([0, 0, 0, -723, 64078]\) \(-27995042/1171875\) \(-1749600000000\) \([2]\) \(512\) \(1.0293\)  
360.b4 360a2 \([0, 0, 0, -183, -182]\) \(3631696/2025\) \(377913600\) \([2, 2]\) \(128\) \(0.33616\)  
360.b5 360a1 \([0, 0, 0, -138, -623]\) \(24918016/45\) \(524880\) \([2]\) \(64\) \(-0.010416\) \(\Gamma_0(N)\)-optimal
360.b6 360a4 \([0, 0, 0, 717, -1442]\) \(54607676/32805\) \(-24488801280\) \([2]\) \(256\) \(0.68273\)