Properties

Label 1254.f
Number of curves $6$
Conductor $1254$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1254.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1254.f1 1254g5 \([1, 1, 1, -148049, -21987223]\) \(358872624127382648977/5938169721462\) \(5938169721462\) \([2]\) \(6144\) \(1.5823\)  
1254.f2 1254g3 \([1, 1, 1, -9539, -324259]\) \(95992014075197617/11235515171364\) \(11235515171364\) \([2, 2]\) \(3072\) \(1.2357\)  
1254.f3 1254g2 \([1, 1, 1, -2319, 36741]\) \(1379233073341297/183927761424\) \(183927761424\) \([2, 4]\) \(1536\) \(0.88916\)  
1254.f4 1254g1 \([1, 1, 1, -2239, 39845]\) \(1241361053832817/27447552\) \(27447552\) \([4]\) \(768\) \(0.54259\) \(\Gamma_0(N)\)-optimal
1254.f5 1254g4 \([1, 1, 1, 3621, 200685]\) \(5250513632788943/20176472892708\) \(-20176472892708\) \([4]\) \(3072\) \(1.2357\)  
1254.f6 1254g6 \([1, 1, 1, 13451, -1620895]\) \(269144439804255023/1298611008739638\) \(-1298611008739638\) \([2]\) \(6144\) \(1.5823\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 1254.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} + q^{8} + q^{9} - 2 q^{10} - q^{11} - q^{12} - 2 q^{13} + 2 q^{15} + q^{16} + 2 q^{17} + q^{18} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.