Properties

Label 78078.bf
Number of curves $4$
Conductor $78078$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 78078.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78078.bf1 78078bf4 \([1, 0, 1, -2388481, -1420891834]\) \(312196988566716625/25367712678\) \(122445103863584502\) \([2]\) \(1327104\) \(2.3240\)  
78078.bf2 78078bf3 \([1, 0, 1, -139091, -25370278]\) \(-61653281712625/21875235228\) \(-105587582275627452\) \([2]\) \(663552\) \(1.9775\)  
78078.bf3 78078bf2 \([1, 0, 1, -61351, 2928434]\) \(5290763640625/2291573592\) \(11060988038027928\) \([2]\) \(442368\) \(1.7747\)  
78078.bf4 78078bf1 \([1, 0, 1, 13009, 340706]\) \(50447927375/39517632\) \(-190744061796288\) \([2]\) \(221184\) \(1.4282\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 78078.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 78078.bf do not have complex multiplication.

Modular form 78078.2.a.bf

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