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Properties

Label 162240.dh
Number of curves $8$
Conductor $162240$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dh1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 162240.dh have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 162240.dh do not have complex multiplication.

Modular form 162240.2.a.dh

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

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 16 & 16 & 8 \\ 2 & 1 & 2 & 2 & 4 & 8 & 8 & 4 \\ 4 & 2 & 1 & 4 & 8 & 16 & 16 & 8 \\ 4 & 2 & 4 & 1 & 2 & 4 & 4 & 2 \\ 8 & 4 & 8 & 2 & 1 & 2 & 2 & 4 \\ 16 & 8 & 16 & 4 & 2 & 1 & 4 & 8 \\ 16 & 8 & 16 & 4 & 2 & 4 & 1 & 8 \\ 8 & 4 & 8 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 162240.dh

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162240.dh1 162240hc8 \([0, -1, 0, -1406080225, -20293326089375]\) \(242970740812818720001/24375\) \(30842151075840000\) \([2]\) \(33030144\) \(3.5116\)  
162240.dh2 162240hc6 \([0, -1, 0, -87880225, -317059649375]\) \(59319456301170001/594140625\) \(751777432473600000000\) \([2, 2]\) \(16515072\) \(3.1650\)  
162240.dh3 162240hc7 \([0, -1, 0, -85771105, -333003331103]\) \(-55150149867714721/5950927734375\) \(-7529822040000000000000000\) \([2]\) \(33030144\) \(3.5116\)  
162240.dh4 162240hc4 \([0, -1, 0, -5624545, -4701930143]\) \(15551989015681/1445900625\) \(1829525559667752960000\) \([2, 2]\) \(8257536\) \(2.8185\)  
162240.dh5 162240hc2 \([0, -1, 0, -1244065, 452142625]\) \(168288035761/27720225\) \(35074927889488281600\) \([2, 2]\) \(4128768\) \(2.4719\)  
162240.dh6 162240hc1 \([0, -1, 0, -1189985, 500025057]\) \(147281603041/5265\) \(6661904632381440\) \([2]\) \(2064384\) \(2.1253\) \(\Gamma_0(N)\)-optimal
162240.dh7 162240hc3 \([0, -1, 0, 2271135, 2540874465]\) \(1023887723039/2798036865\) \(-3540409259737424855040\) \([2]\) \(8257536\) \(2.8185\)  
162240.dh8 162240hc5 \([0, -1, 0, 6543455, -22274955743]\) \(24487529386319/183539412225\) \(-232235908931869743513600\) \([2]\) \(16515072\) \(3.1650\)