Properties

Label 3990.bb
Number of curves $6$
Conductor $3990$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 3990.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3990.bb1 3990bb5 \([1, 0, 0, -355515, -81603333]\) \(4969327007303723277361/1123462695162150\) \(1123462695162150\) \([2]\) \(61440\) \(1.8802\)  
3990.bb2 3990bb3 \([1, 0, 0, -24765, -966483]\) \(1679731262160129361/570261564022500\) \(570261564022500\) \([2, 2]\) \(30720\) \(1.5336\)  
3990.bb3 3990bb2 \([1, 0, 0, -10185, 383625]\) \(116844823575501841/3760263939600\) \(3760263939600\) \([2, 4]\) \(15360\) \(1.1870\)  
3990.bb4 3990bb1 \([1, 0, 0, -10105, 390137]\) \(114113060120923921/124104960\) \(124104960\) \([4]\) \(7680\) \(0.84046\) \(\Gamma_0(N)\)-optimal
3990.bb5 3990bb4 \([1, 0, 0, 3115, 1317285]\) \(3342636501165359/751262567039460\) \(-751262567039460\) \([4]\) \(30720\) \(1.5336\)  
3990.bb6 3990bb6 \([1, 0, 0, 72705, -6678225]\) \(42502666283088696719/43898058864843750\) \(-43898058864843750\) \([2]\) \(61440\) \(1.8802\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3990.bb have rank \(0\).

Complex multiplication

The elliptic curves in class 3990.bb do not have complex multiplication.

Modular form 3990.2.a.bb

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} + 6 q^{13} - q^{14} + q^{15} + q^{16} + 2 q^{17} + q^{18} - q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.