Properties

 Label 570k Number of curves $4$ Conductor $570$ CM no Rank $0$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

Elliptic curves in class 570k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
570.k3 570k1 $$[1, 0, 0, -25871, 1614201]$$ $$-1914980734749238129/20440940544000$$ $$-20440940544000$$ $$$$ $$2880$$ $$1.3705$$ $$\Gamma_0(N)$$-optimal
570.k2 570k2 $$[1, 0, 0, -414991, 102863225]$$ $$7903870428425797297009/886464000000$$ $$886464000000$$ $$$$ $$5760$$ $$1.7171$$
570.k4 570k3 $$[1, 0, 0, 85489, 8420985]$$ $$69096190760262356111/70568821500000000$$ $$-70568821500000000$$ $$$$ $$8640$$ $$1.9198$$
570.k1 570k4 $$[1, 0, 0, -463231, 77449961]$$ $$10993009831928446009969/3767761230468750000$$ $$3767761230468750000$$ $$$$ $$17280$$ $$2.2664$$

Rank

sage: E.rank()

The elliptic curves in class 570k have rank $$0$$.

Complex multiplication

The elliptic curves in class 570k do not have complex multiplication.

Modular form570.2.a.k

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels. 