Properties

Label 1001.b
Number of curves $4$
Conductor $1001$
CM no
Rank $0$
Graph

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

Elliptic curves in class 1001.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1001.b1 1001b3 \([1, -1, 1, -9916, -377564]\) \(107818231938348177/4463459\) \(4463459\) \([2]\) \(608\) \(0.76223\)  
1001.b2 1001b4 \([1, -1, 1, -1006, 2552]\) \(112489728522417/62811265517\) \(62811265517\) \([2]\) \(608\) \(0.76223\)  
1001.b3 1001b2 \([1, -1, 1, -621, -5764]\) \(26444947540257/169338169\) \(169338169\) \([2, 2]\) \(304\) \(0.41565\)  
1001.b4 1001b1 \([1, -1, 1, -16, -198]\) \(-426957777/17320303\) \(-17320303\) \([4]\) \(152\) \(0.069081\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1001.b have rank \(0\).

Complex multiplication

The elliptic curves in class 1001.b do not have complex multiplication.

Modular form 1001.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - 2 q^{5} - q^{7} + 3 q^{8} - 3 q^{9} + 2 q^{10} + q^{11} - q^{13} + q^{14} - q^{16} - 2 q^{17} + 3 q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.