Properties

Label 429a
Number of curves $2$
Conductor $429$
CM no
Rank $1$
Graph

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

Elliptic curves in class 429a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
429.a2 429a1 \([1, 1, 1, 2, 2]\) \(857375/1287\) \(-1287\) \([2]\) \(16\) \(-0.71794\) \(\Gamma_0(N)\)-optimal
429.a1 429a2 \([1, 1, 1, -13, 8]\) \(244140625/61347\) \(61347\) \([2]\) \(32\) \(-0.37136\)  

Rank

sage: E.rank()
 

The elliptic curves in class 429a have rank \(1\).

Complex multiplication

The elliptic curves in class 429a do not have complex multiplication.

Modular form 429.2.a.a

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