# Properties

 Label 107712ek Number of curves $2$ Conductor $107712$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ek1")

sage: E.isogeny_class()

## Elliptic curves in class 107712ek

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
107712.eb2 107712ek1 [0, 0, 0, -6924, 179728] [2] 147456 $$\Gamma_0(N)$$-optimal
107712.eb1 107712ek2 [0, 0, 0, -104844, 13066000] [2] 294912

## Rank

sage: E.rank()

The elliptic curves in class 107712ek have rank $$0$$.

## Complex multiplication

The elliptic curves in class 107712ek do not have complex multiplication.

## Modular form 107712.2.a.ek

sage: E.q_eigenform(10)

$$q + 2q^{5} + 2q^{7} + q^{11} - 4q^{13} - q^{17} + 2q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.