Properties

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

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

Elliptic curves in class 7728.o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7728.o1 7728r5 \([0, 1, 0, -1266144, 547941492]\) \(54804145548726848737/637608031452\) \(2611642496827392\) \([4]\) \(98304\) \(2.1091\)  
7728.o2 7728r3 \([0, 1, 0, -283424, -58169868]\) \(614716917569296417/19093020912\) \(78205013655552\) \([2]\) \(49152\) \(1.7625\)  
7728.o3 7728r4 \([0, 1, 0, -81184, 8073716]\) \(14447092394873377/1439452851984\) \(5895998881726464\) \([2, 4]\) \(49152\) \(1.7625\)  
7728.o4 7728r2 \([0, 1, 0, -18464, -832524]\) \(169967019783457/26337394944\) \(107877969690624\) \([2, 2]\) \(24576\) \(1.4160\)  
7728.o5 7728r1 \([0, 1, 0, 2016, -70668]\) \(221115865823/664731648\) \(-2722740830208\) \([2]\) \(12288\) \(1.0694\) \(\Gamma_0(N)\)-optimal
7728.o6 7728r6 \([0, 1, 0, 100256, 39208820]\) \(27207619911317663/177609314617308\) \(-727487752672493568\) \([4]\) \(98304\) \(2.1091\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 7728.o do not have complex multiplication.

Modular form 7728.2.a.o

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