Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("t1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 7728.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7728.t1 7728p2 \([0, 1, 0, -4004232, -3085271820]\) \(1733490909744055732873/99355964553216\) \(406962030809972736\) \([2]\) \(202752\) \(2.4426\)  
7728.t2 7728p1 \([0, 1, 0, -235912, -54035212]\) \(-354499561600764553/101902222098432\) \(-417391501715177472\) \([2]\) \(101376\) \(2.0961\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 7728.2.a.t

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.