Properties

Label 16744.f
Number of curves $4$
Conductor $16744$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 16744.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16744.f1 16744d4 \([0, 0, 0, -44651, 3631574]\) \(9614292367656708/2093\) \(2143232\) \([2]\) \(16384\) \(1.0392\)  
16744.f2 16744d3 \([0, 0, 0, -3251, 36766]\) \(3710860803108/1577224103\) \(1615077481472\) \([2]\) \(16384\) \(1.0392\)  
16744.f3 16744d2 \([0, 0, 0, -2791, 56730]\) \(9392111857872/4380649\) \(1121446144\) \([2, 2]\) \(8192\) \(0.69262\)  
16744.f4 16744d1 \([0, 0, 0, -146, 1185]\) \(-21511084032/25465531\) \(-407448496\) \([4]\) \(4096\) \(0.34605\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 16744.f have rank \(1\).

Complex multiplication

The elliptic curves in class 16744.f do not have complex multiplication.

Modular form 16744.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.