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Properties

Label 10051b
Number of curves $3$
Conductor $10051$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 10051b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10051.c3 10051b1 \([0, 1, 1, 353, 230]\) \(32768/19\) \(-2812681891\) \([]\) \(4224\) \(0.50257\) \(\Gamma_0(N)\)-optimal
10051.c2 10051b2 \([0, 1, 1, -4937, 140415]\) \(-89915392/6859\) \(-1015378162651\) \([]\) \(12672\) \(1.0519\)  
10051.c1 10051b3 \([0, 1, 1, -406977, 99796080]\) \(-50357871050752/19\) \(-2812681891\) \([]\) \(38016\) \(1.6012\)  

Rank

sage: E.rank()
 

The elliptic curves in class 10051b have rank \(2\).

Complex multiplication

The elliptic curves in class 10051b do not have complex multiplication.

Modular form 10051.2.a.b

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.