Properties

Label 122034.f
Number of curves $2$
Conductor $122034$
CM no
Rank $1$
Graph

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

Elliptic curves in class 122034.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
122034.f1 122034g1 \([1, 1, 1, -360835162, -2639154712345]\) \(-821938895581650775417/282039076306944\) \(-1782871395340807183712256\) \([]\) \(39118464\) \(3.6247\) \(\Gamma_0(N)\)-optimal
122034.f2 122034g2 \([1, 1, 1, 208037423, -9905181771985]\) \(157520606341736640023/6796261691190411264\) \(-42961637526025314587362983936\) \([]\) \(117355392\) \(4.1740\)  

Rank

sage: E.rank()
 

The elliptic curves in class 122034.f have rank \(1\).

Complex multiplication

The elliptic curves in class 122034.f do not have complex multiplication.

Modular form 122034.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} + q^{7} + q^{8} + q^{9} - 3 q^{10} - q^{11} - q^{12} - 4 q^{13} + q^{14} + 3 q^{15} + q^{16} - 3 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.