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Properties

Label 10005i
Number of curves $2$
Conductor $10005$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 10005i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10005.h2 10005i1 \([0, 1, 1, -124021, 14938111]\) \(210966209738334797824/25153051046653125\) \(25153051046653125\) \([3]\) \(64800\) \(1.8780\) \(\Gamma_0(N)\)-optimal
10005.h1 10005i2 \([0, 1, 1, -2379061, -1411043480]\) \(1489157481162281146384384/2616603057861328125\) \(2616603057861328125\) \([]\) \(194400\) \(2.4273\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 10005.2.a.i

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.