Properties

 Label 68544bf Number of curves $4$ Conductor $68544$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bf1")

sage: E.isogeny_class()

Elliptic curves in class 68544bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68544.y4 68544bf1 $$[0, 0, 0, 564, -1393040]$$ $$103823/4386816$$ $$-838333592764416$$ $$[2]$$ $$294912$$ $$1.5424$$ $$\Gamma_0(N)$$-optimal
68544.y3 68544bf2 $$[0, 0, 0, -183756, -29778320]$$ $$3590714269297/73410624$$ $$14028988716417024$$ $$[2, 2]$$ $$589824$$ $$1.8890$$
68544.y2 68544bf3 $$[0, 0, 0, -391116, 49764976]$$ $$34623662831857/14438442312$$ $$2759229294627520512$$ $$[2]$$ $$1179648$$ $$2.2356$$
68544.y1 68544bf4 $$[0, 0, 0, -2925516, -1925979536]$$ $$14489843500598257/6246072$$ $$1193642947510272$$ $$[2]$$ $$1179648$$ $$2.2356$$

Rank

sage: E.rank()

The elliptic curves in class 68544bf have rank $$0$$.

Complex multiplication

The elliptic curves in class 68544bf do not have complex multiplication.

Modular form 68544.2.a.bf

sage: E.q_eigenform(10)

$$q - 2q^{5} - q^{7} + 6q^{13} - q^{17} + O(q^{20})$$

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.