Properties

Label 67600cf
Number of curves $4$
Conductor $67600$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cf1")
 
E.isogeny_class()
 

Elliptic curves in class 67600cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67600.u2 67600cf1 \([0, 1, 0, -2198408, -1247916812]\) \(3803721481/26000\) \(8031810176000000000\) \([2]\) \(2322432\) \(2.4615\) \(\Gamma_0(N)\)-optimal
67600.u3 67600cf2 \([0, 1, 0, -846408, -2764860812]\) \(-217081801/10562500\) \(-3262922884000000000000\) \([2]\) \(4644864\) \(2.8081\)  
67600.u1 67600cf3 \([0, 1, 0, -14028408, 19407263188]\) \(988345570681/44994560\) \(13899529418178560000000\) \([2]\) \(6967296\) \(3.0108\)  
67600.u4 67600cf4 \([0, 1, 0, 7603592, 73876639188]\) \(157376536199/7722894400\) \(-2385723916542054400000000\) \([2]\) \(13934592\) \(3.3574\)  

Rank

sage: E.rank()
 

The elliptic curves in class 67600cf have rank \(0\).

Complex multiplication

The elliptic curves in class 67600cf do not have complex multiplication.

Modular form 67600.2.a.cf

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.