Properties

Label 7728.b
Number of curves $4$
Conductor $7728$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 7728.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7728.b1 7728l4 \([0, -1, 0, -2861584, 1864147264]\) \(632678989847546725777/80515134\) \(329789988864\) \([4]\) \(92160\) \(2.0701\)  
7728.b2 7728l3 \([0, -1, 0, -204624, 20243520]\) \(231331938231569617/90942310746882\) \(372499704819228672\) \([2]\) \(92160\) \(2.0701\)  
7728.b3 7728l2 \([0, -1, 0, -178864, 29166784]\) \(154502321244119857/55101928644\) \(225697499725824\) \([2, 2]\) \(46080\) \(1.7236\)  
7728.b4 7728l1 \([0, -1, 0, -9584, 592320]\) \(-23771111713777/22848457968\) \(-93587283836928\) \([2]\) \(23040\) \(1.3770\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 7728.2.a.b

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.