# Properties

 Label 13680.bk Number of curves $4$ Conductor $13680$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("13680.bk1")

sage: E.isogeny_class()

## Elliptic curves in class 13680.bk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
13680.bk1 13680br3 [0, 0, 0, -89427, -10278254]  49152
13680.bk2 13680br2 [0, 0, 0, -7347, -51086] [2, 2] 24576
13680.bk3 13680br1 [0, 0, 0, -4467, 114226]  12288 $$\Gamma_0(N)$$-optimal
13680.bk4 13680br4 [0, 0, 0, 28653, -403886]  49152

## Rank

sage: E.rank()

The elliptic curves in class 13680.bk have rank $$1$$.

## Modular form 13680.2.a.bk

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 