Properties

Label 162240.cd
Number of curves $4$
Conductor $162240$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cd1")
 
E.isogeny_class()
 

Elliptic curves in class 162240.cd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162240.cd1 162240bz4 \([0, -1, 0, -324705, -71108415]\) \(23937672968/45\) \(7117419479040\) \([2]\) \(1179648\) \(1.7214\)  
162240.cd2 162240bz3 \([0, -1, 0, -54305, 3440865]\) \(111980168/32805\) \(5188598800220160\) \([2]\) \(1179648\) \(1.7214\)  
162240.cd3 162240bz2 \([0, -1, 0, -20505, -1081575]\) \(48228544/2025\) \(40035484569600\) \([2, 2]\) \(589824\) \(1.3748\)  
162240.cd4 162240bz1 \([0, -1, 0, 620, -63350]\) \(85184/5625\) \(-1737651240000\) \([2]\) \(294912\) \(1.0282\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 162240.cd have rank \(1\).

Complex multiplication

The elliptic curves in class 162240.cd do not have complex multiplication.

Modular form 162240.2.a.cd

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} - 4 q^{7} + q^{9} - q^{15} - 6 q^{17} + O(q^{20})\) 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.