Properties

Label 431970dz
Number of curves $2$
Conductor $431970$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands for: SageMath
sage: E = EllipticCurve("dz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 431970dz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
431970.dz2 431970dz1 \([1, 1, 1, 9859, -445741]\) \(59822347031/83966400\) \(-148751599550400\) \([2]\) \(1612800\) \(1.4051\) \(\Gamma_0(N)\)-optimal*
431970.dz1 431970dz2 \([1, 1, 1, -62741, -4482301]\) \(15417797707369/4080067320\) \(7228088141486520\) \([2]\) \(3225600\) \(1.7516\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 431970dz1.

Rank

sage: E.rank()
 

The elliptic curves in class 431970dz have rank \(1\).

Complex multiplication

The elliptic curves in class 431970dz do not have complex multiplication.

Modular form 431970.2.a.dz

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