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Properties

Label 1005a
Number of curves $2$
Conductor $1005$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 1005a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1005.b1 1005a1 \([1, 1, 0, -297, -1944]\) \(2912566550041/254390625\) \(254390625\) \([2]\) \(360\) \(0.35342\) \(\Gamma_0(N)\)-optimal
1005.b2 1005a2 \([1, 1, 0, 328, -8319]\) \(3883959939959/33133870125\) \(-33133870125\) \([2]\) \(720\) \(0.69999\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1005a have rank \(0\).

Complex multiplication

The elliptic curves in class 1005a do not have complex multiplication.

Modular form 1005.2.a.a

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