Properties

Label 10005e
Number of curves $4$
Conductor $10005$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 10005e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10005.d4 10005e1 \([1, 1, 1, -5250, -144690]\) \(16003198512756001/488525390625\) \(488525390625\) \([4]\) \(12288\) \(1.0189\) \(\Gamma_0(N)\)-optimal
10005.d2 10005e2 \([1, 1, 1, -83375, -9300940]\) \(64096096056024006001/62562515625\) \(62562515625\) \([2, 2]\) \(24576\) \(1.3654\)  
10005.d1 10005e3 \([1, 1, 1, -1334000, -593592940]\) \(262537424941059264096001/250125\) \(250125\) \([2]\) \(49152\) \(1.7120\)  
10005.d3 10005e4 \([1, 1, 1, -82750, -9446440]\) \(-62665433378363916001/2004003001000125\) \(-2004003001000125\) \([2]\) \(49152\) \(1.7120\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 10005.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} + 3 q^{8} + q^{9} - q^{10} + q^{12} + 6 q^{13} - q^{15} - q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.