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Properties

Label 42978.r
Number of curves $2$
Conductor $42978$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 42978.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42978.r1 42978v1 \([1, 0, 0, -803575, 284546441]\) \(-57385609248046791394801/1784804944651399008\) \(-1784804944651399008\) \([5]\) \(1800000\) \(2.2789\) \(\Gamma_0(N)\)-optimal
42978.r2 42978v2 \([1, 0, 0, 3910355, -13981592329]\) \(6612600178387895114470319/88280159906925363790998\) \(-88280159906925363790998\) \([]\) \(9000000\) \(3.0837\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42978.r have rank \(2\).

Complex multiplication

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

Modular form 42978.2.a.r

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.