Properties

Label 27200.h
Number of curves $4$
Conductor $27200$
CM no
Rank $1$
Graph

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

Elliptic curves in class 27200.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27200.h1 27200o3 \([0, 1, 0, -6669633, -1306311137]\) \(8010684753304969/4456448000000\) \(18253611008000000000000\) \([2]\) \(2211840\) \(2.9619\)  
27200.h2 27200o1 \([0, 1, 0, -4085633, 3177200863]\) \(1841373668746009/31443200\) \(128791347200000000\) \([2]\) \(737280\) \(2.4126\) \(\Gamma_0(N)\)-optimal
27200.h3 27200o2 \([0, 1, 0, -3957633, 3385712863]\) \(-1673672305534489/241375690000\) \(-988674826240000000000\) \([2]\) \(1474560\) \(2.7592\)  
27200.h4 27200o4 \([0, 1, 0, 26098367, -10317511137]\) \(479958568556831351/289000000000000\) \(-1183744000000000000000000\) \([2]\) \(4423680\) \(3.3085\)  

Rank

sage: E.rank()
 

The elliptic curves in class 27200.h have rank \(1\).

Complex multiplication

The elliptic curves in class 27200.h do not have complex multiplication.

Modular form 27200.2.a.h

sage: E.q_eigenform(10)
 
\(q - 2 q^{3} - 2 q^{7} + q^{9} - 6 q^{11} + 2 q^{13} - q^{17} - 8 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.