# Properties

 Label 188598x Number of curves 6 Conductor 188598 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("188598.c1")

sage: E.isogeny_class()

## Elliptic curves in class 188598x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
188598.c5 188598x1 [1, 1, 0, -62904, -5658048] [2] 1204224 $$\Gamma_0(N)$$-optimal
188598.c4 188598x2 [1, 1, 0, -210824, 30641520] [2, 2] 2408448
188598.c2 188598x3 [1, 1, 0, -3206204, 2208282780] [2, 2] 4816896
188598.c6 188598x4 [1, 1, 0, 417836, 179131012] [2] 4816896
188598.c1 188598x5 [1, 1, 0, -51298694, 141397567338] [2] 9633792
188598.c3 188598x6 [1, 1, 0, -3039794, 2447946462] [2] 9633792

## Rank

sage: E.rank()

The elliptic curves in class 188598x have rank $$1$$.

## Modular form 188598.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} - q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} - 2q^{13} - 2q^{15} + q^{16} + q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.