# Properties

 Label 4114b Number of curves 4 Conductor 4114 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("4114.a1")

sage: E.isogeny_class()

## Elliptic curves in class 4114b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4114.a4 4114b1 [1, 0, 1, -366, -1696]  2160 $$\Gamma_0(N)$$-optimal
4114.a3 4114b2 [1, 0, 1, -5206, -144960]  4320
4114.a2 4114b3 [1, 0, 1, -12466, 534576]  6480
4114.a1 4114b4 [1, 0, 1, -13676, 424224]  12960

## Rank

sage: E.rank()

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

## Modular form4114.2.a.a

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. 