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Properties

Label 10005j
Number of curves $2$
Conductor $10005$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 10005j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10005.m1 10005j1 \([1, 0, 1, -374, 2747]\) \(5763259856089/450225\) \(450225\) \([2]\) \(1728\) \(0.13203\) \(\Gamma_0(N)\)-optimal
10005.m2 10005j2 \([1, 0, 1, -349, 3137]\) \(-4681768588489/1621620405\) \(-1621620405\) \([2]\) \(3456\) \(0.47860\)  

Rank

sage: E.rank()
 

The elliptic curves in class 10005j have rank \(1\).

Complex multiplication

The elliptic curves in class 10005j do not have complex multiplication.

Modular form 10005.2.a.j

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.