Properties

Label 285.a
Number of curves $2$
Conductor $285$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 285.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
285.a1 285a2 \([1, 0, 0, -76, -19]\) \(48587168449/28048275\) \(28048275\) \([2]\) \(80\) \(0.11864\)  
285.a2 285a1 \([1, 0, 0, 19, 0]\) \(756058031/438615\) \(-438615\) \([2]\) \(40\) \(-0.22793\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 285.a have rank \(1\).

Complex multiplication

The elliptic curves in class 285.a do not have complex multiplication.

Modular form 285.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2q^{7} + 3q^{8} + q^{9} + q^{10} - 6q^{11} - q^{12} + 2q^{14} - q^{15} - q^{16} - 6q^{17} - q^{18} + q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.