# Properties

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

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

sage: E.isogeny_class()

## Elliptic curves in class 13680.bk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13680.bk1 13680br3 $$[0, 0, 0, -89427, -10278254]$$ $$26487576322129/44531250$$ $$132969600000000$$ $$$$ $$49152$$ $$1.6054$$
13680.bk2 13680br2 $$[0, 0, 0, -7347, -51086]$$ $$14688124849/8122500$$ $$24253655040000$$ $$[2, 2]$$ $$24576$$ $$1.2588$$
13680.bk3 13680br1 $$[0, 0, 0, -4467, 114226]$$ $$3301293169/22800$$ $$68080435200$$ $$$$ $$12288$$ $$0.91222$$ $$\Gamma_0(N)$$-optimal
13680.bk4 13680br4 $$[0, 0, 0, 28653, -403886]$$ $$871257511151/527800050$$ $$-1576002504499200$$ $$$$ $$49152$$ $$1.6054$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 13680.bk do not have complex multiplication.

## 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. 