Properties

Label 16575.f
Number of curves $4$
Conductor $16575$
CM no
Rank $1$
Graph

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

Elliptic curves in class 16575.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16575.f1 16575a3 \([1, 1, 0, -390150, -93961125]\) \(420339554066191969/244298925\) \(3817170703125\) \([2]\) \(98304\) \(1.7389\)  
16575.f2 16575a2 \([1, 1, 0, -24525, -1458000]\) \(104413920565969/2472575625\) \(38633994140625\) \([2, 2]\) \(49152\) \(1.3923\)  
16575.f3 16575a1 \([1, 1, 0, -3400, 41875]\) \(278317173889/109245825\) \(1706966015625\) \([2]\) \(24576\) \(1.0457\) \(\Gamma_0(N)\)-optimal
16575.f4 16575a4 \([1, 1, 0, 3100, -4524375]\) \(210751100351/566398828125\) \(-8849981689453125\) \([2]\) \(98304\) \(1.7389\)  

Rank

sage: E.rank()
 

The elliptic curves in class 16575.f have rank \(1\).

Complex multiplication

The elliptic curves in class 16575.f do not have complex multiplication.

Modular form 16575.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} - q^{4} - q^{6} - 4 q^{7} - 3 q^{8} + q^{9} + q^{12} - q^{13} - 4 q^{14} - q^{16} - q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.