# Properties

 Label 12789.k Number of curves $6$ Conductor $12789$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("12789.k1")

sage: E.isogeny_class()

## Elliptic curves in class 12789.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
12789.k1 12789f4 [1, -1, 0, -90239193, 329967079434] [2] 589824
12789.k2 12789f5 [1, -1, 0, -18728838, -25356051351] [2] 1179648
12789.k3 12789f3 [1, -1, 0, -5748003, 4949006040] [2, 2] 589824
12789.k4 12789f2 [1, -1, 0, -5639958, 5156776575] [2, 2] 294912
12789.k5 12789f1 [1, -1, 0, -345753, 83869344] [2] 147456 $$\Gamma_0(N)$$-optimal
12789.k6 12789f6 [1, -1, 0, 5504112, 21955452651] [2] 1179648

## Rank

sage: E.rank()

The elliptic curves in class 12789.k have rank $$1$$.

## Modular form 12789.2.a.k

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - q^{16} + 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.