Properties

Label 162450.d
Number of curves $4$
Conductor $162450$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 162450.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162450.d1 162450dd3 \([1, -1, 0, -246925692, 1493535273966]\) \(3107086841064961/570\) \(305452733311406250\) \([2]\) \(26542080\) \(3.1908\)  
162450.d2 162450dd4 \([1, -1, 0, -17871192, 15475826466]\) \(1177918188481/488703750\) \(261887537222866933593750\) \([2]\) \(26542080\) \(3.1908\)  
162450.d3 162450dd2 \([1, -1, 0, -15434442, 23334345216]\) \(758800078561/324900\) \(174108057987501562500\) \([2, 2]\) \(13271040\) \(2.8442\)  
162450.d4 162450dd1 \([1, -1, 0, -813942, 482503716]\) \(-111284641/123120\) \(-65977790395263750000\) \([2]\) \(6635520\) \(2.4976\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 162450.d have rank \(1\).

Complex multiplication

The elliptic curves in class 162450.d do not have complex multiplication.

Modular form 162450.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 4 q^{7} - q^{8} + 4 q^{11} - 2 q^{13} + 4 q^{14} + q^{16} - 2 q^{17} + O(q^{20})\) 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.