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Properties

Label 49686ct
Number of curves $2$
Conductor $49686$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ct1")
 
E.isogeny_class()
 

Elliptic curves in class 49686ct

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
49686.ct2 49686ct1 \([1, 0, 0, 3968, -117664]\) \(596183/864\) \(-10013041505376\) \([]\) \(129600\) \(1.1801\) \(\Gamma_0(N)\)-optimal
49686.ct1 49686ct2 \([1, 0, 0, -120247, -16141399]\) \(-16591834777/98304\) \(-1139261611278336\) \([]\) \(388800\) \(1.7294\)  

Rank

sage: E.rank()
 

The elliptic curves in class 49686ct have rank \(1\).

Complex multiplication

The elliptic curves in class 49686ct do not have complex multiplication.

Modular form 49686.2.a.ct

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