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Properties

Label 42978x
Number of curves $3$
Conductor $42978$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 42978x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42978.v2 42978x1 \([1, 0, 0, -1493, 25569]\) \(-368062283004625/72186536448\) \(-72186536448\) \([3]\) \(38880\) \(0.80676\) \(\Gamma_0(N)\)-optimal
42978.v3 42978x2 \([1, 0, 0, 10387, -132327]\) \(123934293043859375/79385028425352\) \(-79385028425352\) \([3]\) \(116640\) \(1.3561\)  
42978.v1 42978x3 \([1, 0, 0, -174863, -29029845]\) \(-591313293127508640625/21499590336237858\) \(-21499590336237858\) \([]\) \(349920\) \(1.9054\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42978x have rank \(1\).

Complex multiplication

The elliptic curves in class 42978x do not have complex multiplication.

Modular form 42978.2.a.x

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.