# Properties

 Label 4114b Number of curves $4$ Conductor $4114$ CM no Rank $1$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 4114b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4114.a4 4114b1 $$[1, 0, 1, -366, -1696]$$ $$3048625/1088$$ $$1927458368$$ $$$$ $$2160$$ $$0.48256$$ $$\Gamma_0(N)$$-optimal
4114.a3 4114b2 $$[1, 0, 1, -5206, -144960]$$ $$8805624625/2312$$ $$4095849032$$ $$$$ $$4320$$ $$0.82913$$
4114.a2 4114b3 $$[1, 0, 1, -12466, 534576]$$ $$120920208625/19652$$ $$34814716772$$ $$$$ $$6480$$ $$1.0319$$
4114.a1 4114b4 $$[1, 0, 1, -13676, 424224]$$ $$159661140625/48275138$$ $$85522351750418$$ $$$$ $$12960$$ $$1.3784$$

## Rank

sage: E.rank()

The elliptic curves in class 4114b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4114b do not have complex multiplication.

## Modular form4114.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} + 4q^{7} - q^{8} + q^{9} - 2q^{12} - 2q^{13} - 4q^{14} + q^{16} + q^{17} - q^{18} + 4q^{19} + O(q^{20})$$ ## 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 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. 