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Properties

Label 6384.i
Number of curves $6$
Conductor $6384$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([0, -1, 0, -156587368, 754246418800]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 6384.i have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 6384.i do not have complex multiplication.

Modular form 6384.2.a.i

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} - q^{7} + q^{9} - 6 q^{11} - 4 q^{13} - q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 6384.i

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6384.i1 6384q6 \([0, -1, 0, -156587368, 754246418800]\) \(103665426767620308239307625/5961940992\) \(24420110303232\) \([2]\) \(497664\) \(2.9554\)  
6384.i2 6384q5 \([0, -1, 0, -9786728, 11787501936]\) \(25309080274342544331625/191933498523648\) \(786159609952862208\) \([2]\) \(248832\) \(2.6088\)  
6384.i3 6384q4 \([0, -1, 0, -1934968, 1033192048]\) \(195607431345044517625/752875610010048\) \(3083778498601156608\) \([2]\) \(165888\) \(2.4061\)  
6384.i4 6384q3 \([0, -1, 0, -178808, -834960]\) \(154357248921765625/89242711068672\) \(365538144537280512\) \([2]\) \(82944\) \(2.0595\)  
6384.i5 6384q2 \([0, -1, 0, -127048, -16288016]\) \(55369510069623625/3916046302812\) \(16040125656317952\) \([2]\) \(55296\) \(1.8568\)  
6384.i6 6384q1 \([0, -1, 0, -124808, -16929552]\) \(52492168638015625/293197968\) \(1200938876928\) \([2]\) \(27648\) \(1.5102\) \(\Gamma_0(N)\)-optimal