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Properties

Label 49686z
Number of curves $2$
Conductor $49686$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("z1")
 
E.isogeny_class()
 

Elliptic curves in class 49686z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
49686.ba1 49686z1 \([1, 0, 1, -182355, 56990182]\) \(-24100657/36504\) \(-1015745459648728536\) \([]\) \(1185408\) \(2.1468\) \(\Gamma_0(N)\)-optimal
49686.ba2 49686z2 \([1, 0, 1, 1556655, -1142926718]\) \(14991903983/28960854\) \(-805852946472981547686\) \([]\) \(3556224\) \(2.6961\)  

Rank

sage: E.rank()
 

The elliptic curves in class 49686z have rank \(0\).

Complex multiplication

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

Modular form 49686.2.a.z

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