# Properties

 Label 31680x Number of curves $4$ Conductor $31680$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

## Elliptic curves in class 31680x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31680.bk4 31680x1 $$[0, 0, 0, 7872, 987752]$$ $$72268906496/606436875$$ $$-452702701440000$$ $$$$ $$73728$$ $$1.4936$$ $$\Gamma_0(N)$$-optimal
31680.bk3 31680x2 $$[0, 0, 0, -113628, 13575152]$$ $$13584145739344/1195803675$$ $$14282602562764800$$ $$$$ $$147456$$ $$1.8401$$
31680.bk2 31680x3 $$[0, 0, 0, -562368, 162451208]$$ $$-26348629355659264/24169921875$$ $$-18042750000000000$$ $$$$ $$221184$$ $$2.0429$$
31680.bk1 31680x4 $$[0, 0, 0, -8999868, 10392076208]$$ $$6749703004355978704/5671875$$ $$67744512000000$$ $$$$ $$442368$$ $$2.3894$$

## Rank

sage: E.rank()

The elliptic curves in class 31680x have rank $$1$$.

## Complex multiplication

The elliptic curves in class 31680x do not have complex multiplication.

## Modular form 31680.2.a.x

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 