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Properties

Label 92400hf
Number of curves $8$
Conductor $92400$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("hf1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 92400hf have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 92400hf do not have complex multiplication.

Modular form 92400.2.a.hf

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 92400hf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
92400.hu6 92400hf1 \([0, 1, 0, -114613408, -472320044812]\) \(2601656892010848045529/56330588160\) \(3605157642240000000\) \([2]\) \(7962624\) \(3.0877\) \(\Gamma_0(N)\)-optimal
92400.hu5 92400hf2 \([0, 1, 0, -114741408, -471212332812]\) \(2610383204210122997209/12104550027662400\) \(774691201770393600000000\) \([2, 2]\) \(15925248\) \(3.4342\)  
92400.hu4 92400hf3 \([0, 1, 0, -122299408, -405367928812]\) \(3160944030998056790089/720291785342976000\) \(46098674261950464000000000\) \([2]\) \(23887872\) \(3.6370\)  
92400.hu7 92400hf4 \([0, 1, 0, -56421408, -949786252812]\) \(-310366976336070130009/5909282337130963560\) \(-378194069576381667840000000\) \([4]\) \(31850496\) \(3.7808\)  
92400.hu3 92400hf5 \([0, 1, 0, -175109408, 78257203188]\) \(9278380528613437145689/5328033205714065000\) \(340994125165700160000000000\) \([2]\) \(31850496\) \(3.7808\)  
92400.hu2 92400hf6 \([0, 1, 0, -646587408, 5981508487188]\) \(467116778179943012100169/28800309694464000000\) \(1843219820445696000000000000\) \([2, 2]\) \(47775744\) \(3.9835\)  
92400.hu8 92400hf7 \([0, 1, 0, 505412592, 24977988487188]\) \(223090928422700449019831/4340371122724101696000\) \(-277783751854342508544000000000\) \([4]\) \(95551488\) \(4.3301\)  
92400.hu1 92400hf8 \([0, 1, 0, -10187195408, 395753507719188]\) \(1826870018430810435423307849/7641104625000000000\) \(489030696000000000000000000\) \([2]\) \(95551488\) \(4.3301\)