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Properties

Label 109200gf
Number of curves $4$
Conductor $109200$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gf1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 109200gf have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 109200gf do not have complex multiplication.

Modular form 109200.2.a.gf

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 109200gf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
109200.gm3 109200gf1 \([0, 1, 0, -127374408, 439555171188]\) \(3571003510905229697089/762141946675200000\) \(48777084587212800000000000\) \([2]\) \(26542080\) \(3.6434\) \(\Gamma_0(N)\)-optimal
109200.gm2 109200gf2 \([0, 1, 0, -651662408, -6018624412812]\) \(478202393398338853167169/32244226560000000000\) \(2063630499840000000000000000\) \([2, 2]\) \(53084160\) \(3.9900\)  
109200.gm4 109200gf3 \([0, 1, 0, 559729592, -25829729180812]\) \(303025056761573589385151/4678857421875000000000\) \(-299446875000000000000000000000\) \([2]\) \(106168320\) \(4.3366\)  
109200.gm1 109200gf4 \([0, 1, 0, -10251662408, -399522624412812]\) \(1861772567578966373029167169/9401133413380800000\) \(601672538456371200000000000\) \([2]\) \(106168320\) \(4.3366\)