Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("bz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51600.bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.bz1 51600x4 \([0, 1, 0, -229408, -42368812]\) \(41725476313778/17415\) \(557280000000\) \([2]\) \(294912\) \(1.5984\)  
51600.bz2 51600x2 \([0, 1, 0, -14408, -658812]\) \(20674973956/416025\) \(6656400000000\) \([2, 2]\) \(147456\) \(1.2518\)  
51600.bz3 51600x1 \([0, 1, 0, -1908, 16188]\) \(192143824/80625\) \(322500000000\) \([2]\) \(73728\) \(0.90525\) \(\Gamma_0(N)\)-optimal
51600.bz4 51600x3 \([0, 1, 0, 592, -1948812]\) \(715822/51282015\) \(-1641024480000000\) \([2]\) \(294912\) \(1.5984\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 51600.2.a.bz

sage: E.q_eigenform(10)
 
\(q + q^{3} - 4q^{7} + q^{9} - 4q^{11} + 2q^{13} - 6q^{17} + 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.