Properties

 Label 405600ca Number of curves $4$ Conductor $405600$ CM no Rank $0$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("ca1")

sage: E.isogeny_class()

Elliptic curves in class 405600ca

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.ca3 405600ca1 $$[0, -1, 0, -280258, -53163488]$$ $$504358336/38025$$ $$183539412225000000$$ $$[2, 2]$$ $$3096576$$ $$2.0578$$ $$\Gamma_0(N)$$-optimal*
405600.ca2 405600ca2 $$[0, -1, 0, -914008, 273851512]$$ $$2186875592/428415$$ $$16543019021880000000$$ $$$$ $$6193152$$ $$2.4044$$ $$\Gamma_0(N)$$-optimal*
405600.ca4 405600ca3 $$[0, -1, 0, 268992, -236612988]$$ $$55742968/658125$$ $$-25413149385000000000$$ $$$$ $$6193152$$ $$2.4044$$
405600.ca1 405600ca4 $$[0, -1, 0, -4399633, -3550512863]$$ $$30488290624/195$$ $$60238576320000000$$ $$$$ $$6193152$$ $$2.4044$$
*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 405600ca1.

Rank

sage: E.rank()

The elliptic curves in class 405600ca have rank $$0$$.

Complex multiplication

The elliptic curves in class 405600ca do not have complex multiplication.

Modular form 405600.2.a.ca

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} - 2 q^{17} - 4 q^{19} + O(q^{20})$$ 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels. 