Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("i1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 576i

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

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 576.2.a.i

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

Isogeny matrix

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.