# Properties

 Label 111540.bj Number of curves $4$ Conductor $111540$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bj1")

sage: E.isogeny_class()

## Elliptic curves in class 111540.bj

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
111540.bj1 111540bi4 $$[0, 1, 0, -42249380, -105714969372]$$ $$6749703004355978704/5671875$$ $$7008526668000000$$ $$$$ $$3981312$$ $$2.7760$$
111540.bj2 111540bi3 $$[0, 1, 0, -2640005, -1653219372]$$ $$-26348629355659264/24169921875$$ $$-1866617542968750000$$ $$$$ $$1990656$$ $$2.4295$$
111540.bj3 111540bi2 $$[0, 1, 0, -533420, -138254700]$$ $$13584145739344/1195803675$$ $$1477610480825107200$$ $$$$ $$1327104$$ $$2.2267$$
111540.bj4 111540bi1 $$[0, 1, 0, 36955, -10034400]$$ $$72268906496/606436875$$ $$-46834479458910000$$ $$$$ $$663552$$ $$1.8801$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 111540.bj have rank $$0$$.

## Complex multiplication

The elliptic curves in class 111540.bj do not have complex multiplication.

## Modular form 111540.2.a.bj

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} - 2q^{7} + q^{9} - q^{11} + q^{15} - 2q^{19} + O(q^{20})$$ ## 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 LMFDB 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 LMFDB labels. 