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Properties

Label 576.b
Number of curves $6$
Conductor $576$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -13836, -626416]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -13836, -626416]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -13836, -626416]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 576.b have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 576.b do not have complex multiplication.

Modular form 576.2.a.b

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - 2 q^{5} - 4 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 576.b

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
576.b1 576i5 \([0, 0, 0, -13836, -626416]\) \(3065617154/9\) \(859963392\) \([2]\) \(512\) \(0.94368\)  
576.b2 576i4 \([0, 0, 0, -2316, 42896]\) \(28756228/3\) \(143327232\) \([2]\) \(256\) \(0.59710\)  
576.b3 576i3 \([0, 0, 0, -876, -9520]\) \(1556068/81\) \(3869835264\) \([2, 2]\) \(256\) \(0.59710\)  
576.b4 576i2 \([0, 0, 0, -156, 560]\) \(35152/9\) \(107495424\) \([2, 2]\) \(128\) \(0.25053\)  
576.b5 576i1 \([0, 0, 0, 24, 56]\) \(2048/3\) \(-2239488\) \([2]\) \(64\) \(-0.096046\) \(\Gamma_0(N)\)-optimal
576.b6 576i6 \([0, 0, 0, 564, -37744]\) \(207646/6561\) \(-626913312768\) \([2]\) \(512\) \(0.94368\)