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Properties

Label 67830.a
Number of curves $2$
Conductor $67830$
CM no
Rank $1$
Graph

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

Elliptic curves in class 67830.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67830.a1 67830b2 \([1, 1, 0, -175273, -28198667]\) \(595487492723207345689/2876601913794000\) \(2876601913794000\) \([2]\) \(731136\) \(1.8149\)  
67830.a2 67830b1 \([1, 1, 0, -5273, -896667]\) \(-16219061115665689/336246876000000\) \(-336246876000000\) \([2]\) \(365568\) \(1.4683\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 67830.a have rank \(1\).

Complex multiplication

The elliptic curves in class 67830.a do not have complex multiplication.

Modular form 67830.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + q^{10} - 6 q^{11} - q^{12} + 2 q^{13} + q^{14} + q^{15} + q^{16} + q^{17} - q^{18} + q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.