Properties

Label 327990.bf
Number of curves $4$
Conductor $327990$
CM no
Rank $0$
Graph

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

Elliptic curves in class 327990.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
327990.bf1 327990bf4 \([1, 1, 1, -733837695, -7651781299995]\) \(73474353581350183614361/576510977802240\) \(342922174409285678039040\) \([2]\) \(108864000\) \(3.6890\)  
327990.bf2 327990bf3 \([1, 1, 1, -44890495, -124895350555]\) \(-16818951115904497561/1592332281446400\) \(-947156375785454331494400\) \([2]\) \(54432000\) \(3.3425\)  
327990.bf3 327990bf2 \([1, 1, 1, -13458120, 709765545]\) \(453198971846635561/261896250564000\) \(155781997517926603044000\) \([2]\) \(36288000\) \(3.1397\)  
327990.bf4 327990bf1 \([1, 1, 1, 3361880, 90789545]\) \(7064514799444439/4094064000000\) \(-2435244744866544000000\) \([2]\) \(18144000\) \(2.7932\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 327990.bf have rank \(0\).

Complex multiplication

The elliptic curves in class 327990.bf do not have complex multiplication.

Modular form 327990.2.a.bf

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