Properties

Label 8085.f
Number of curves $6$
Conductor $8085$
CM no
Rank $0$
Graph

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

Elliptic curves in class 8085.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8085.f1 8085g5 \([1, 1, 1, -149410801, 702882354698]\) \(3135316978843283198764801/571725\) \(67262874525\) \([2]\) \(368640\) \(2.8721\)  
8085.f2 8085g4 \([1, 1, 1, -9338176, 10979616248]\) \(765458482133960722801/326869475625\) \(38455866937805625\) \([2, 2]\) \(184320\) \(2.5255\)  
8085.f3 8085g6 \([1, 1, 1, -9291871, 11093952554]\) \(-754127868744065783521/15825714261328125\) \(-1861879457130992578125\) \([2]\) \(368640\) \(2.8721\)  
8085.f4 8085g3 \([1, 1, 1, -1246806, -283100196]\) \(1821931919215868881/761147600816295\) \(89548254088436290455\) \([2]\) \(184320\) \(2.5255\)  
8085.f5 8085g2 \([1, 1, 1, -586531, 169584344]\) \(189674274234120481/3859869269025\) \(454109759631522225\) \([2, 2]\) \(92160\) \(2.1789\)  
8085.f6 8085g1 \([1, 1, 1, 1714, 7934618]\) \(4733169839/231139696095\) \(-27193354105880655\) \([2]\) \(46080\) \(1.8324\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8085.f have rank \(0\).

Complex multiplication

The elliptic curves in class 8085.f do not have complex multiplication.

Modular form 8085.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.