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Properties

Label 1254.e
Number of curves $2$
Conductor $1254$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 1254.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1254.e1 1254d1 \([1, 0, 1, -206, 896]\) \(960044289625/195182592\) \(195182592\) \([2]\) \(448\) \(0.30645\) \(\Gamma_0(N)\)-optimal
1254.e2 1254d2 \([1, 0, 1, 434, 5504]\) \(9070486526375/18165704832\) \(-18165704832\) \([2]\) \(896\) \(0.65303\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1254.e have rank \(1\).

Complex multiplication

The elliptic curves in class 1254.e do not have complex multiplication.

Modular form 1254.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - 2 q^{7} - q^{8} + q^{9} - q^{11} + q^{12} - 4 q^{13} + 2 q^{14} + q^{16} + 2 q^{17} - q^{18} + q^{19} + O(q^{20})\) 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.