# Properties

 Label 98192q Number of curves 4 Conductor 98192 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("98192.f1")

sage: E.isogeny_class()

## Elliptic curves in class 98192q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
98192.f4 98192q1 [0, 1, 0, -17448, 543412] [2] 331776 $$\Gamma_0(N)$$-optimal
98192.f3 98192q2 [0, 1, 0, -248488, 47583156] [2] 663552
98192.f2 98192q3 [0, 1, 0, -595048, -176849100] [2] 995328
98192.f1 98192q4 [0, 1, 0, -652808, -140506508] [2] 1990656

## Rank

sage: E.rank()

The elliptic curves in class 98192q have rank $$1$$.

## Modular form 98192.2.a.f

sage: E.q_eigenform(10)

$$q - 2q^{3} + 4q^{7} + q^{9} - 6q^{11} - 2q^{13} - q^{17} + O(q^{20})$$

## 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.