Properties

Label 76664.b
Number of curves $4$
Conductor $76664$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 76664.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76664.b1 76664b4 \([0, 0, 0, -409331, 100799470]\) \(1443468546/7\) \(36782253799424\) \([2]\) \(414720\) \(1.8034\)  
76664.b2 76664b3 \([0, 0, 0, -80771, -6990114]\) \(11090466/2401\) \(12616313053202432\) \([2]\) \(414720\) \(1.8034\)  
76664.b3 76664b2 \([0, 0, 0, -26011, 1519590]\) \(740772/49\) \(128737888297984\) \([2, 2]\) \(207360\) \(1.4569\)  
76664.b4 76664b1 \([0, 0, 0, 1369, 101306]\) \(432/7\) \(-4597781724928\) \([2]\) \(103680\) \(1.1103\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 76664.b have rank \(1\).

Complex multiplication

The elliptic curves in class 76664.b do not have complex multiplication.

Modular form 76664.2.a.b

sage: E.q_eigenform(10)
 
\(q - 2q^{5} - q^{7} - 3q^{9} - 4q^{11} - 2q^{13} + 6q^{17} - 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.