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Properties

Label 73920.gp
Number of curves $8$
Conductor $73920$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 73920.gp have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 73920.gp do not have complex multiplication.

Modular form 73920.2.a.gp

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{5} - q^{7} + q^{9} - q^{11} - 2 q^{13} + q^{15} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 4 & 6 & 12 \\ 4 & 1 & 2 & 3 & 12 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 12 & 3 & 6 & 1 & 4 & 12 & 2 & 4 \\ 3 & 12 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 73920.gp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
73920.gp1 73920hf7 \([0, 1, 0, -340736065, 2420777609663]\) \(16689299266861680229173649/2396798250\) \(628306280448000\) \([2]\) \(7962624\) \(3.1621\)  
73920.gp2 73920hf8 \([0, 1, 0, -21856065, 35724585663]\) \(4404531606962679693649/444872222400201750\) \(116620583868878487552000\) \([2]\) \(7962624\) \(3.1621\)  
73920.gp3 73920hf6 \([0, 1, 0, -21296065, 37819097663]\) \(4074571110566294433649/48828650062500\) \(12800137641984000000\) \([2, 2]\) \(3981312\) \(2.8156\)  
73920.gp4 73920hf5 \([0, 1, 0, -4800705, -4042199745]\) \(46676570542430835889/106752955783320\) \(27984646840862638080\) \([2]\) \(2654208\) \(2.6128\)  
73920.gp5 73920hf4 \([0, 1, 0, -4211905, 3310668095]\) \(31522423139920199089/164434491947880\) \(43105515457185054720\) \([2]\) \(2654208\) \(2.6128\)  
73920.gp6 73920hf3 \([0, 1, 0, -1296065, 623097663]\) \(-918468938249433649/109183593750000\) \(-28621824000000000000\) \([2]\) \(1990656\) \(2.4690\)  
73920.gp7 73920hf2 \([0, 1, 0, -410305, -12690625]\) \(29141055407581489/16604321025600\) \(4352723130934886400\) \([2, 2]\) \(1327104\) \(2.2663\)  
73920.gp8 73920hf1 \([0, 1, 0, 101695, -1529025]\) \(443688652450511/260789760000\) \(-68364470845440000\) \([2]\) \(663552\) \(1.9197\) \(\Gamma_0(N)\)-optimal