Properties

Label 7728.o
Number of curves $6$
Conductor $7728$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("7728.o1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 7728.o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
7728.o1 7728r5 [0, 1, 0, -1266144, 547941492] [4] 98304  
7728.o2 7728r3 [0, 1, 0, -283424, -58169868] [2] 49152  
7728.o3 7728r4 [0, 1, 0, -81184, 8073716] [2, 4] 49152  
7728.o4 7728r2 [0, 1, 0, -18464, -832524] [2, 2] 24576  
7728.o5 7728r1 [0, 1, 0, 2016, -70668] [2] 12288 \(\Gamma_0(N)\)-optimal
7728.o6 7728r6 [0, 1, 0, 100256, 39208820] [4] 98304  

Rank

sage: E.rank()
 

The elliptic curves in class 7728.o have rank \(0\).

Modular form 7728.2.a.o

sage: E.q_eigenform(10)
 
\( q + q^{3} - 2q^{5} - q^{7} + q^{9} + 4q^{11} - 2q^{13} - 2q^{15} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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.