Properties

Label 21450.h
Number of curves $4$
Conductor $21450$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 21450.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21450.h1 21450a4 \([1, 1, 0, -4010650, -3093105500]\) \(456612868287073618849/12544848030000\) \(196013250468750000\) \([2]\) \(589824\) \(2.4215\)  
21450.h2 21450a3 \([1, 1, 0, -1118650, 411242500]\) \(9908022260084596129/1047363281250000\) \(16365051269531250000\) \([2]\) \(589824\) \(2.4215\)  
21450.h3 21450a2 \([1, 1, 0, -260650, -44355500]\) \(125337052492018849/18404100000000\) \(287564062500000000\) \([2, 2]\) \(294912\) \(2.0749\)  
21450.h4 21450a1 \([1, 1, 0, 27350, -3747500]\) \(144794100308831/474439680000\) \(-7413120000000000\) \([2]\) \(147456\) \(1.7283\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 21450.h have rank \(1\).

Complex multiplication

The elliptic curves in class 21450.h do not have complex multiplication.

Modular form 21450.2.a.h

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