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Properties

Label 405042.q
Number of curves $2$
Conductor $405042$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("q1")
 
E.isogeny_class()
 

Elliptic curves in class 405042.q

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405042.q1 405042q1 \([1, 1, 0, -71485, -7386071]\) \(858729462625/38148\) \(1794706268388\) \([2]\) \(1548288\) \(1.4286\) \(\Gamma_0(N)\)-optimal
405042.q2 405042q2 \([1, 1, 0, -67875, -8160777]\) \(-735091890625/181908738\) \(-8558056840808178\) \([2]\) \(3096576\) \(1.7752\)  

Rank

sage: E.rank()
 

The elliptic curves in class 405042.q have rank \(1\).

Complex multiplication

The elliptic curves in class 405042.q do not have complex multiplication.

Modular form 405042.2.a.q

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