Properties

 Label 236992bp Number of curves $4$ Conductor $236992$ CM no Rank $1$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("bp1")

sage: E.isogeny_class()

Elliptic curves in class 236992bp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
236992.bp4 236992bp1 $$[0, 0, 0, 2116, 194672]$$ $$432/7$$ $$-16977940037632$$ $$$$ $$405504$$ $$1.2191$$ $$\Gamma_0(N)$$-optimal
236992.bp3 236992bp2 $$[0, 0, 0, -40204, 2920080]$$ $$740772/49$$ $$475382321053696$$ $$[2, 2]$$ $$811008$$ $$1.5657$$
236992.bp1 236992bp3 $$[0, 0, 0, -632684, 193698640]$$ $$1443468546/7$$ $$135823520301056$$ $$$$ $$1622016$$ $$1.9123$$
236992.bp2 236992bp4 $$[0, 0, 0, -124844, -13432368]$$ $$11090466/2401$$ $$46587467463262208$$ $$$$ $$1622016$$ $$1.9123$$

Rank

sage: E.rank()

The elliptic curves in class 236992bp have rank $$1$$.

Complex multiplication

The elliptic curves in class 236992bp do not have complex multiplication.

Modular form 236992.2.a.bp

sage: E.q_eigenform(10)

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels. 