Properties

Label 21175.p
Number of curves $2$
Conductor $21175$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 21175.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21175.p1 21175bl2 \([1, 1, 1, -157968, 24099956]\) \(1968634623437/5929\) \(1312948146125\) \([2]\) \(99840\) \(1.5531\)  
21175.p2 21175bl1 \([1, 1, 1, -9743, 383956]\) \(-461889917/26411\) \(-5848587196375\) \([2]\) \(49920\) \(1.2065\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 21175.p have rank \(0\).

Complex multiplication

The elliptic curves in class 21175.p do not have complex multiplication.

Modular form 21175.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{2} + 2 q^{3} - q^{4} - 2 q^{6} + q^{7} + 3 q^{8} + q^{9} - 2 q^{12} + 6 q^{13} - q^{14} - q^{16} + 6 q^{17} - q^{18} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.