# Properties

 Label 8712k Number of curves $4$ Conductor $8712$ CM no Rank $1$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 8712k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8712.f3 8712k1 $$[0, 0, 0, -13431, -577654]$$ $$810448/33$$ $$10910321402112$$ $$$$ $$15360$$ $$1.2675$$ $$\Gamma_0(N)$$-optimal
8712.f2 8712k2 $$[0, 0, 0, -35211, 1770230]$$ $$3650692/1089$$ $$1440162425078784$$ $$[2, 2]$$ $$30720$$ $$1.6141$$
8712.f1 8712k3 $$[0, 0, 0, -514371, 141972446]$$ $$5690357426/891$$ $$2356629422856192$$ $$$$ $$61440$$ $$1.9607$$
8712.f4 8712k4 $$[0, 0, 0, 95469, 11832590]$$ $$36382894/43923$$ $$-116173102289688576$$ $$$$ $$61440$$ $$1.9607$$

## Rank

sage: E.rank()

The elliptic curves in class 8712k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 8712k do not have complex multiplication.

## Modular form8712.2.a.k

sage: E.q_eigenform(10)

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