Properties

Label 1288f
Number of curves $2$
Conductor $1288$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1288f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1288.c2 1288f1 \([0, 1, 0, 4192, 16816]\) \(7953970437500/4703287687\) \(-4816166591488\) \([2]\) \(1920\) \(1.1229\) \(\Gamma_0(N)\)-optimal
1288.c1 1288f2 \([0, 1, 0, -16968, 118384]\) \(263822189935250/149429406721\) \(306031424964608\) \([2]\) \(3840\) \(1.4695\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1288f have rank \(1\).

Complex multiplication

The elliptic curves in class 1288f do not have complex multiplication.

Modular form 1288.2.a.f

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} + q^{7} + q^{9} - 6 q^{17} + 6 q^{19} + O(q^{20})\)  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 Cremona 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 Cremona labels.