Properties

Label 187200.hz
Number of curves $4$
Conductor $187200$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("hz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 187200.hz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
187200.hz1 187200ml4 \([0, 0, 0, -194700, -26894000]\) \(2186875592/428415\) \(159905041920000000\) \([2]\) \(1179648\) \(2.0178\)  
187200.hz2 187200ml2 \([0, 0, 0, -59700, 5236000]\) \(504358336/38025\) \(1774094400000000\) \([2, 2]\) \(589824\) \(1.6712\)  
187200.hz3 187200ml1 \([0, 0, 0, -58575, 5456500]\) \(30488290624/195\) \(142155000000\) \([2]\) \(294912\) \(1.3247\) \(\Gamma_0(N)\)-optimal
187200.hz4 187200ml3 \([0, 0, 0, 57300, 23254000]\) \(55742968/658125\) \(-245643840000000000\) \([2]\) \(1179648\) \(2.0178\)  

Rank

sage: E.rank()
 

The elliptic curves in class 187200.hz have rank \(2\).

Complex multiplication

The elliptic curves in class 187200.hz do not have complex multiplication.

Modular form 187200.2.a.hz

sage: E.q_eigenform(10)
 
\(q - q^{13} + 2 q^{17} - 4 q^{19} + O(q^{20})\)  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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.