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Properties

Label 6370.z
Number of curves $6$
Conductor $6370$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("z1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 6370.z have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6370.z do not have complex multiplication.

Modular form 6370.2.a.z

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + 2 q^{3} + q^{4} + q^{5} + 2 q^{6} + q^{8} + q^{9} + q^{10} + 2 q^{12} - q^{13} + 2 q^{15} + q^{16} + q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 6 & 18 \\ 2 & 1 & 6 & 18 & 3 & 9 \\ 3 & 6 & 1 & 3 & 2 & 6 \\ 9 & 18 & 3 & 1 & 6 & 2 \\ 6 & 3 & 2 & 6 & 1 & 3 \\ 18 & 9 & 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 6370.z

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6370.z1 6370v6 \([1, 1, 1, -25923255, 50767188527]\) \(16375858190544687071329/9025573730468750\) \(1061849723815917968750\) \([2]\) \(497664\) \(2.9825\)  
6370.z2 6370v5 \([1, 1, 1, -25919825, 50781305035]\) \(16369358802802724130049/4976562500\) \(585487601562500\) \([2]\) \(248832\) \(2.6359\)  
6370.z3 6370v4 \([1, 1, 1, -997445, -301506605]\) \(932829715460155969/206949435875000\) \(24347394181257875000\) \([2]\) \(165888\) \(2.4332\)  
6370.z4 6370v2 \([1, 1, 1, -936685, -349320413]\) \(772531501373731009/15142400\) \(1781488217600\) \([2]\) \(55296\) \(1.8839\)  
6370.z5 6370v3 \([1, 1, 1, -325165, 67171747]\) \(32318182904349889/2067798824000\) \(243274463844776000\) \([2]\) \(82944\) \(2.0866\)  
6370.z6 6370v1 \([1, 1, 1, -58605, -5464285]\) \(189208196468929/834928640\) \(98228519567360\) \([2]\) \(27648\) \(1.5373\) \(\Gamma_0(N)\)-optimal