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Properties

Label 42978u
Number of curves $2$
Conductor $42978$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 42978u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
42978.x1 42978u1 \([1, 0, 0, -162, -588]\) \(470366406433/129621648\) \(129621648\) \([2]\) \(17152\) \(0.26419\) \(\Gamma_0(N)\)-optimal
42978.x2 42978u2 \([1, 0, 0, 418, -3720]\) \(8075838390047/10764183924\) \(-10764183924\) \([2]\) \(34304\) \(0.61077\)  

Rank

sage: E.rank()
 

The elliptic curves in class 42978u have rank \(0\).

Complex multiplication

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

Modular form 42978.2.a.u

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.