Properties

 Label 414400s Number of curves $2$ Conductor $414400$ CM no Rank $0$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("s1")

sage: E.isogeny_class()

Elliptic curves in class 414400s

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414400.s2 414400s1 $$[0, 1, 0, 1567, 1055263]$$ $$415292/469567$$ $$-480836608000000$$ $$$$ $$1769472$$ $$1.4962$$ $$\Gamma_0(N)$$-optimal*
414400.s1 414400s2 $$[0, 1, 0, -146433, 21035263]$$ $$169556172914/4353013$$ $$8914970624000000$$ $$$$ $$3538944$$ $$1.8428$$ $$\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 414400s1.

Rank

sage: E.rank()

The elliptic curves in class 414400s have rank $$0$$.

Complex multiplication

The elliptic curves in class 414400s do not have complex multiplication.

Modular form 414400.2.a.s

sage: E.q_eigenform(10)

$$q - 2 q^{3} - q^{7} + q^{9} + 4 q^{11} - 6 q^{13} + 4 q^{17} + 4 q^{19} + O(q^{20})$$ 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels. 