Properties

Label 21175r
Number of curves $3$
Conductor $21175$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("r1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 21175r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21175.u2 21175r1 \([0, -1, 1, -4033, -108032]\) \(-262144/35\) \(-968822421875\) \([]\) \(21600\) \(1.0325\) \(\Gamma_0(N)\)-optimal
21175.u3 21175r2 \([0, -1, 1, 26217, 270093]\) \(71991296/42875\) \(-1186807466796875\) \([]\) \(64800\) \(1.5818\)  
21175.u1 21175r3 \([0, -1, 1, -397283, 100957218]\) \(-250523582464/13671875\) \(-378446258544921875\) \([]\) \(194400\) \(2.1311\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21175r have rank \(1\).

Complex multiplication

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

Modular form 21175.2.a.r

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.