Properties

Label 51600.dn
Number of curves $4$
Conductor $51600$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("dn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51600.dn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.dn1 51600dh3 \([0, 1, 0, -13722408, 8494795188]\) \(4465136636671380769/2096375976562500\) \(134168062500000000000000\) \([2]\) \(4976640\) \(3.1321\)  
51600.dn2 51600dh1 \([0, 1, 0, -7026408, -7170820812]\) \(599437478278595809/33854760000\) \(2166704640000000000\) \([2]\) \(1658880\) \(2.5828\) \(\Gamma_0(N)\)-optimal
51600.dn3 51600dh2 \([0, 1, 0, -6626408, -8022820812]\) \(-502780379797811809/143268096832200\) \(-9169158197260800000000\) \([2]\) \(3317760\) \(2.9293\)  
51600.dn4 51600dh4 \([0, 1, 0, 48777592, 64369795188]\) \(200541749524551119231/144008551960031250\) \(-9216547325442000000000000\) \([2]\) \(9953280\) \(3.4786\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51600.dn have rank \(0\).

Complex multiplication

The elliptic curves in class 51600.dn do not have complex multiplication.

Modular form 51600.2.a.dn

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.