# Properties

 Label 8712k Number of curves 4 Conductor 8712 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("8712.f1")

sage: E.isogeny_class()

## Elliptic curves in class 8712k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
8712.f3 8712k1 [0, 0, 0, -13431, -577654]  15360 $$\Gamma_0(N)$$-optimal
8712.f2 8712k2 [0, 0, 0, -35211, 1770230] [2, 2] 30720
8712.f1 8712k3 [0, 0, 0, -514371, 141972446]  61440
8712.f4 8712k4 [0, 0, 0, 95469, 11832590]  61440

## Rank

sage: E.rank()

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

## Modular form8712.2.a.f

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. 