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Properties

Label 576d
Number of curves $6$
Conductor $576$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("d1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 576d have rank \(0\).

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.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 576.2.a.d

Copy content sage:E.q_eigenform(10)
 
\(q - 2 q^{5} + 4 q^{11} + 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 576d

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
576.d5 576d1 \([0, 0, 0, 24, -56]\) \(2048/3\) \(-2239488\) \([2]\) \(64\) \(-0.096046\) \(\Gamma_0(N)\)-optimal
576.d4 576d2 \([0, 0, 0, -156, -560]\) \(35152/9\) \(107495424\) \([2, 2]\) \(128\) \(0.25053\)  
576.d2 576d3 \([0, 0, 0, -2316, -42896]\) \(28756228/3\) \(143327232\) \([2]\) \(256\) \(0.59710\)  
576.d3 576d4 \([0, 0, 0, -876, 9520]\) \(1556068/81\) \(3869835264\) \([2, 2]\) \(256\) \(0.59710\)  
576.d1 576d5 \([0, 0, 0, -13836, 626416]\) \(3065617154/9\) \(859963392\) \([2]\) \(512\) \(0.94368\)  
576.d6 576d6 \([0, 0, 0, 564, 37744]\) \(207646/6561\) \(-626913312768\) \([2]\) \(512\) \(0.94368\)