Properties

 Label 168b Number of curves $4$ Conductor $168$ CM no Rank $0$ Graph Related objects

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

sage: E.isogeny_class()

Elliptic curves in class 168b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
168.a4 168b1 $$[0, -1, 0, -7, 52]$$ $$-2725888/64827$$ $$-1037232$$ $$$$ $$24$$ $$-0.16496$$ $$\Gamma_0(N)$$-optimal
168.a3 168b2 $$[0, -1, 0, -252, 1620]$$ $$6940769488/35721$$ $$9144576$$ $$[2, 2]$$ $$48$$ $$0.18162$$
168.a2 168b3 $$[0, -1, 0, -392, -228]$$ $$6522128932/3720087$$ $$3809369088$$ $$$$ $$96$$ $$0.52819$$
168.a1 168b4 $$[0, -1, 0, -4032, 99900]$$ $$7080974546692/189$$ $$193536$$ $$$$ $$96$$ $$0.52819$$

Rank

sage: E.rank()

The elliptic curves in class 168b have rank $$0$$.

Complex multiplication

The elliptic curves in class 168b do not have complex multiplication.

Modular form168.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} + 2 q^{5} + q^{7} + q^{9} + 6 q^{13} - 2 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})$$ 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels. 