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Properties

Label 159390.dz
Number of curves $4$
Conductor $159390$
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 159390.dz have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1 + T\)
\(11\)\(1 - T\)
\(23\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 159390.dz do not have complex multiplication.

Modular form 159390.2.a.dz

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} + q^{11} + 2 q^{13} - q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.

Elliptic curves in class 159390.dz

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
159390.dz1 159390k4 \([1, -1, 1, -12050036717, -509129281114891]\) \(265436898662503851515370589836169/17149152760523437500000\) \(12501732362421585937500000\) \([2]\) \(188743680\) \(4.2742\)  
159390.dz2 159390k2 \([1, -1, 1, -754580237, -7922768001739]\) \(65179715853307296723232286089/520784732418538896000000\) \(379652069933114855184000000\) \([2, 2]\) \(94371840\) \(3.9277\)  
159390.dz3 159390k3 \([1, -1, 1, -253640237, -18280804569739]\) \(-2475429904568270179255646089/196606057528071366356412000\) \(-143325815937964026073824348000\) \([2]\) \(188743680\) \(4.2742\)  
159390.dz4 159390k1 \([1, -1, 1, -79922957, 70301588789]\) \(77448107425788419878921609/41892392875786371072000\) \(30539554406448264511488000\) \([4]\) \(47185920\) \(3.5811\) \(\Gamma_0(N)\)-optimal