Properties

Label 24.a
Number of curves $6$
Conductor $24$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 24.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
24.a1 24a5 \([0, -1, 0, -384, -2772]\) \(3065617154/9\) \(18432\) \([2]\) \(4\) \(0.047795\)  
24.a2 24a3 \([0, -1, 0, -64, 220]\) \(28756228/3\) \(3072\) \([4]\) \(2\) \(-0.29878\)  
24.a3 24a2 \([0, -1, 0, -24, -36]\) \(1556068/81\) \(82944\) \([2, 2]\) \(2\) \(-0.29878\)  
24.a4 24a1 \([0, -1, 0, -4, 4]\) \(35152/9\) \(2304\) \([2, 4]\) \(1\) \(-0.64535\) \(\Gamma_0(N)\)-optimal
24.a5 24a4 \([0, -1, 0, 1, 0]\) \(2048/3\) \(-48\) \([4]\) \(2\) \(-0.99193\)  
24.a6 24a6 \([0, -1, 0, 16, -180]\) \(207646/6561\) \(-13436928\) \([2]\) \(4\) \(0.047795\)  

Rank

sage: E.rank()
 

The elliptic curves in class 24.a have rank \(0\).

Complex multiplication

The elliptic curves in class 24.a do not have complex multiplication.

Modular form 24.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2q^{5} + q^{9} + 4q^{11} - 2q^{13} + 2q^{15} + 2q^{17} - 4q^{19} + O(q^{20})\)  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 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

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

The vertices are labelled with LMFDB labels.