# Properties

 Label 8624.k Number of curves $2$ Conductor $8624$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 8624.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8624.k1 8624m2 $$[0, -1, 0, -4328, 109936]$$ $$911871625/10648$$ $$104717713408$$ $$[]$$ $$6912$$ $$0.92682$$
8624.k2 8624m1 $$[0, -1, 0, -408, -2960]$$ $$765625/22$$ $$216358912$$ $$[]$$ $$2304$$ $$0.37751$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 8624.k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 8624.k do not have complex multiplication.

## Modular form8624.2.a.k

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{9} + q^{11} - q^{13} - 6q^{17} - 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 