Properties

Label 7728.a
Number of curves $2$
Conductor $7728$
CM no
Rank $1$
Graph

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

Elliptic curves in class 7728.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7728.a1 7728j2 \([0, -1, 0, -432, 15936]\) \(-2181825073/25039686\) \(-102562553856\) \([]\) \(8640\) \(0.79475\)  
7728.a2 7728j1 \([0, -1, 0, 48, -576]\) \(2924207/34776\) \(-142442496\) \([]\) \(2880\) \(0.24544\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 7728.a have rank \(1\).

Complex multiplication

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

Modular form 7728.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{3} - 3 q^{5} - q^{7} + q^{9} + 5 q^{13} + 3 q^{15} - 8 q^{19} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.