Properties

Label 1248.f
Number of curves $4$
Conductor $1248$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1248.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1248.f1 1248e3 \([0, 1, 0, -3744, 86940]\) \(11339065490696/351\) \(179712\) \([2]\) \(768\) \(0.51272\)  
1248.f2 1248e2 \([0, 1, 0, -369, -513]\) \(1360251712/771147\) \(3158618112\) \([2]\) \(768\) \(0.51272\)  
1248.f3 1248e1 \([0, 1, 0, -234, 1296]\) \(22235451328/123201\) \(7884864\) \([2, 2]\) \(384\) \(0.16614\) \(\Gamma_0(N)\)-optimal
1248.f4 1248e4 \([0, 1, 0, -104, 2856]\) \(-245314376/6908733\) \(-3537271296\) \([4]\) \(768\) \(0.51272\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1248.f have rank \(1\).

Complex multiplication

The elliptic curves in class 1248.f do not have complex multiplication.

Modular form 1248.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} + q^{9} - 4 q^{11} + q^{13} - 2 q^{15} - 6 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.