Properties

Label 84042.f
Number of curves $4$
Conductor $84042$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 84042.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
84042.f1 84042l4 \([1, -1, 0, -103987968, -408126777040]\) \(170586815436843383543017473/2166416\) \(1579317264\) \([2]\) \(4915200\) \(2.7515\)  
84042.f2 84042l3 \([1, -1, 0, -6515808, -6341625136]\) \(41966336340198080824833/442001722607124848\) \(322219255780594014192\) \([2]\) \(4915200\) \(2.7515\)  
84042.f3 84042l2 \([1, -1, 0, -6499248, -6375761920]\) \(41647175116728660507393/4693358285056\) \(3421458189805824\) \([2, 2]\) \(2457600\) \(2.4050\)  
84042.f4 84042l1 \([1, -1, 0, -405168, -100078336]\) \(-10090256344188054273/107965577101312\) \(-78706905706856448\) \([2]\) \(1228800\) \(2.0584\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 84042.f have rank \(0\).

Complex multiplication

The elliptic curves in class 84042.f do not have complex multiplication.

Modular form 84042.2.a.f

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