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Properties

Label 129472.q
Number of curves $6$
Conductor $129472$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("q1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 129472.q have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 129472.q do not have complex multiplication.

Modular form 129472.2.a.q

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{3} - q^{7} + q^{9} + 4 q^{13} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 129472.q

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129472.q1 129472l6 \([0, 1, 0, -50503713, -138161168321]\) \(2251439055699625/25088\) \(158744793860538368\) \([2]\) \(5308416\) \(2.8694\)  
129472.q2 129472l5 \([0, 1, 0, -3153953, -2163187649]\) \(-548347731625/1835008\) \(-11611047779513663488\) \([2]\) \(2654208\) \(2.5229\)  
129472.q3 129472l4 \([0, 1, 0, -656993, -168227585]\) \(4956477625/941192\) \(5955410157174259712\) \([2]\) \(1769472\) \(2.3201\)  
129472.q4 129472l2 \([0, 1, 0, -194593, 32953407]\) \(128787625/98\) \(620096851017728\) \([2]\) \(589824\) \(1.7708\)  
129472.q5 129472l1 \([0, 1, 0, -9633, 733375]\) \(-15625/28\) \(-177170528862208\) \([2]\) \(294912\) \(1.4242\) \(\Gamma_0(N)\)-optimal
129472.q6 129472l3 \([0, 1, 0, 82847, -15376641]\) \(9938375/21952\) \(-138901694627971072\) \([2]\) \(884736\) \(1.9735\)