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Properties

Label 21021n
Number of curves $2$
Conductor $21021$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("n1")
 
E.isogeny_class()
 

Elliptic curves in class 21021n

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21021.f2 21021n1 \([1, 0, 0, 97, -456]\) \(857375/1287\) \(-151414263\) \([2]\) \(6144\) \(0.25502\) \(\Gamma_0(N)\)-optimal
21021.f1 21021n2 \([1, 0, 0, -638, -4719]\) \(244140625/61347\) \(7217413203\) \([2]\) \(12288\) \(0.60159\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21021n have rank \(0\).

Complex multiplication

The elliptic curves in class 21021n do not have complex multiplication.

Modular form 21021.2.a.n

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - q^{6} + 3 q^{8} + q^{9} + q^{11} - q^{12} - q^{13} - q^{16} + 4 q^{17} - q^{18} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.