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Properties

Label 259920.ew
Number of curves $4$
Conductor $259920$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ew1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 259920.ew have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 259920.ew do not have complex multiplication.

Modular form 259920.2.a.ew

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 259920.ew

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259920.ew1 259920ew4 \([0, 0, 0, -24080601387, -918421731577766]\) \(10993009831928446009969/3767761230468750000\) \(529288496282004750000000000000000\) \([2]\) \(1194393600\) \(4.9810\)  
259920.ew2 259920ew2 \([0, 0, 0, -21572893227, -1219581035519654]\) \(7903870428425797297009/886464000000\) \(124528909574707347456000000\) \([2]\) \(398131200\) \(4.4317\)  
259920.ew3 259920ew1 \([0, 0, 0, -1344879147, -19157631147686]\) \(-1914980734749238129/20440940544000\) \(-2871507513701340624715776000\) \([2]\) \(199065600\) \(4.0852\) \(\Gamma_0(N)\)-optimal
259920.ew4 259920ew3 \([0, 0, 0, 4444059093, -99724041277094]\) \(69096190760262356111/70568821500000000\) \(-9913384402939277531136000000000\) \([2]\) \(597196800\) \(4.6345\)