Properties

Label 8085.t
Number of curves $4$
Conductor $8085$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("t1")
 
E.isogeny_class()
 

Elliptic curves in class 8085.t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8085.t1 8085l4 \([1, 1, 0, -100622, 12243519]\) \(957681397954009/31185\) \(3668884065\) \([2]\) \(18432\) \(1.3365\)  
8085.t2 8085l3 \([1, 1, 0, -9972, -61851]\) \(932288503609/527295615\) \(62035801809135\) \([2]\) \(18432\) \(1.3365\)  
8085.t3 8085l2 \([1, 1, 0, -6297, 188784]\) \(234770924809/1334025\) \(156946707225\) \([2, 2]\) \(9216\) \(0.98997\)  
8085.t4 8085l1 \([1, 1, 0, -172, 6259]\) \(-4826809/144375\) \(-16985574375\) \([2]\) \(4608\) \(0.64340\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 8085.2.a.t

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.