Properties

 Label 414736bz Number of curves $2$ Conductor $414736$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bz1")

sage: E.isogeny_class()

Elliptic curves in class 414736bz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414736.bz2 414736bz1 $$[0, -1, 0, 14299752, -97909629712]$$ $$4533086375/60669952$$ $$-4328015978908947381747712$$ $$[2]$$ $$68124672$$ $$3.4080$$ $$\Gamma_0(N)$$-optimal*
414736.bz1 414736bz2 $$[0, -1, 0, -251131288, -1433983312656]$$ $$24553362849625/1755162752$$ $$125208149764842438707904512$$ $$[2]$$ $$136249344$$ $$3.7546$$ $$\Gamma_0(N)$$-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 414736bz1.

Rank

sage: E.rank()

The elliptic curves in class 414736bz have rank $$0$$.

Complex multiplication

The elliptic curves in class 414736bz do not have complex multiplication.

Modular form 414736.2.a.bz

sage: E.q_eigenform(10)

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

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.