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Properties

Label 122304gh
Number of curves $6$
Conductor $122304$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("gh1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 122304gh have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(7\)\(1\)
\(13\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + T + 5 T^{2}\) 1.5.b
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 + 9 T + 29 T^{2}\) 1.29.j
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 122304gh do not have complex multiplication.

Modular form 122304.2.a.gh

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 122304gh

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
122304.u5 122304gh1 \([0, -1, 0, -46909, 3774205]\) \(94757435392/4179357\) \(503497903813632\) \([2]\) \(393216\) \(1.5841\) \(\Gamma_0(N)\)-optimal
122304.u4 122304gh2 \([0, -1, 0, -126289, -12276431]\) \(115562131792/32867289\) \(63353711551463424\) \([2, 2]\) \(786432\) \(1.9307\)  
122304.u6 122304gh3 \([0, -1, 0, 332351, -81347615]\) \(526556774012/674481717\) \(-5200419194361151488\) \([2]\) \(1572864\) \(2.2772\)  
122304.u2 122304gh4 \([0, -1, 0, -1855009, -971716031]\) \(91557481657828/12595401\) \(97113625870270464\) \([2, 2]\) \(1572864\) \(2.2772\)  
122304.u3 122304gh5 \([0, -1, 0, -1690369, -1151404127]\) \(-34639400027234/17130345141\) \(-264158311283885211648\) \([2]\) \(3145728\) \(2.6238\)  
122304.u1 122304gh6 \([0, -1, 0, -29679169, -62223821855]\) \(187491149065688834/3549\) \(54727318044672\) \([2]\) \(3145728\) \(2.6238\)