Properties

Label 12992.f
Number of curves $2$
Conductor $12992$
CM no
Rank $1$
Graph

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

Elliptic curves in class 12992.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12992.f1 12992u2 \([0, 1, 0, -9889, 375231]\) \(408023180713/1421\) \(372506624\) \([2]\) \(12288\) \(0.86409\)  
12992.f2 12992u1 \([0, 1, 0, -609, 5887]\) \(-95443993/5887\) \(-1543241728\) \([2]\) \(6144\) \(0.51751\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 12992.f have rank \(1\).

Complex multiplication

The elliptic curves in class 12992.f do not have complex multiplication.

Modular form 12992.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.