Properties

Label 600d
Number of curves $6$
Conductor $600$
CM no
Rank $0$
Graph

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

Elliptic curves in class 600d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
600.h5 600d1 \([0, 1, 0, 17, 38]\) \(2048/3\) \(-750000\) \([2]\) \(64\) \(-0.18721\) \(\Gamma_0(N)\)-optimal
600.h4 600d2 \([0, 1, 0, -108, 288]\) \(35152/9\) \(36000000\) \([2, 2]\) \(128\) \(0.15937\)  
600.h3 600d3 \([0, 1, 0, -608, -5712]\) \(1556068/81\) \(1296000000\) \([2, 2]\) \(256\) \(0.50594\)  
600.h2 600d4 \([0, 1, 0, -1608, 24288]\) \(28756228/3\) \(48000000\) \([2]\) \(256\) \(0.50594\)  
600.h1 600d5 \([0, 1, 0, -9608, -365712]\) \(3065617154/9\) \(288000000\) \([2]\) \(512\) \(0.85251\)  
600.h6 600d6 \([0, 1, 0, 392, -21712]\) \(207646/6561\) \(-209952000000\) \([2]\) \(512\) \(0.85251\)  

Rank

sage: E.rank()
 

The elliptic curves in class 600d have rank \(0\).

Complex multiplication

The elliptic curves in class 600d do not have complex multiplication.

Modular form 600.2.a.d

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.