# Properties

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

# Related objects

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)

$$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - q^{10} - q^{12} + 2q^{13} - q^{14} + q^{15} + q^{16} - q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.