Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 7728.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7728.p1 7728u2 \([0, 1, 0, -53544, -4183884]\) \(4144806984356137/568114785504\) \(2326998161424384\) \([2]\) \(46080\) \(1.6755\)  
7728.p2 7728u1 \([0, 1, 0, 5336, -344908]\) \(4101378352343/15049939968\) \(-61644554108928\) \([2]\) \(23040\) \(1.3289\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 7728.2.a.p

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.