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Properties

Label 84474.ba
Number of curves $3$
Conductor $84474$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ba1")
 
E.isogeny_class()
 

Elliptic curves in class 84474.ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
84474.ba1 84474q3 \([1, -1, 0, -1492983, -701778083]\) \(-10730978619193/6656\) \(-228277152889344\) \([]\) \(1283040\) \(2.0759\)  
84474.ba2 84474q2 \([1, -1, 0, -14688, -1361912]\) \(-10218313/17576\) \(-602794356848424\) \([]\) \(427680\) \(1.5266\)  
84474.ba3 84474q1 \([1, -1, 0, 1557, 38407]\) \(12167/26\) \(-891707628474\) \([]\) \(142560\) \(0.97730\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 84474.ba have rank \(1\).

Complex multiplication

The elliptic curves in class 84474.ba do not have complex multiplication.

Modular form 84474.2.a.ba

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.