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Properties

Label 162288.dz
Number of curves $4$
Conductor $162288$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dz1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 162288.dz have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(7\)\(1\)
\(23\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 162288.dz do not have complex multiplication.

Modular form 162288.2.a.dz

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

\(\left(\begin{array}{rrrr} 1 & 3 & 2 & 6 \\ 3 & 1 & 6 & 2 \\ 2 & 6 & 1 & 3 \\ 6 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 162288.dz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162288.dz1 162288bu4 \([0, 0, 0, -10700357115, 423969404771626]\) \(385693937170561837203625/2159357734550274048\) \(758578121702296308055768301568\) \([2]\) \(265420800\) \(4.5758\)  
162288.dz2 162288bu2 \([0, 0, 0, -790240395, -8146587987398]\) \(155355156733986861625/8291568305839392\) \(2912811624850990288234217472\) \([2]\) \(88473600\) \(4.0265\)  
162288.dz3 162288bu3 \([0, 0, 0, -295861755, 13971941514538]\) \(-8152944444844179625/235342826399858688\) \(-82675471669216356834956279808\) \([2]\) \(132710400\) \(4.2292\)  
162288.dz4 162288bu1 \([0, 0, 0, 32771445, -508544305094]\) \(11079872671250375/324440155855872\) \(-113975188129356089395249152\) \([2]\) \(44236800\) \(3.6799\) \(\Gamma_0(N)\)-optimal