Properties

 Label 816.b Number of curves $6$ Conductor $816$ CM no Rank $1$ Graph

Related objects

Show commands: SageMath
E = EllipticCurve("b1")

E.isogeny_class()

Elliptic curves in class 816.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
816.b1 816h5 $$[0, -1, 0, -443904, 113984640]$$ $$2361739090258884097/5202$$ $$21307392$$ $$[4]$$ $$3072$$ $$1.5402$$
816.b2 816h3 $$[0, -1, 0, -27744, 1787904]$$ $$576615941610337/27060804$$ $$110841053184$$ $$[2, 4]$$ $$1536$$ $$1.1937$$
816.b3 816h6 $$[0, -1, 0, -26304, 1980288]$$ $$-491411892194497/125563633938$$ $$-514308644610048$$ $$[4]$$ $$3072$$ $$1.5402$$
816.b4 816h2 $$[0, -1, 0, -1824, 25344]$$ $$163936758817/30338064$$ $$124264710144$$ $$[2, 2]$$ $$768$$ $$0.84708$$
816.b5 816h1 $$[0, -1, 0, -544, -4352]$$ $$4354703137/352512$$ $$1443889152$$ $$[2]$$ $$384$$ $$0.50050$$ $$\Gamma_0(N)$$-optimal
816.b6 816h4 $$[0, -1, 0, 3616, 142848]$$ $$1276229915423/2927177028$$ $$-11989717106688$$ $$[2]$$ $$1536$$ $$1.1937$$

Rank

sage: E.rank()

The elliptic curves in class 816.b have rank $$1$$.

Complex multiplication

The elliptic curves in class 816.b do not have complex multiplication.

Modular form816.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} - 2 q^{5} + q^{9} + 4 q^{11} - 2 q^{13} + 2 q^{15} + q^{17} - 4 q^{19} + O(q^{20})$$

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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels.