# Properties

 Label 194145j Number of curves 4 Conductor 194145 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("194145.d1")

sage: E.isogeny_class()

## Elliptic curves in class 194145j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
194145.d3 194145j1 [1, 1, 1, -4661, -117982]  322560 $$\Gamma_0(N)$$-optimal
194145.d2 194145j2 [1, 1, 1, -13906, 481094] [2, 2] 645120
194145.d1 194145j3 [1, 1, 1, -208051, 36436748]  1290240
194145.d4 194145j4 [1, 1, 1, 32319, 3051204]  1290240

## Rank

sage: E.rank()

The elliptic curves in class 194145j have rank $$0$$.

## Modular form 194145.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - q^{7} + 3q^{8} + q^{9} + q^{10} + q^{12} - 6q^{13} + q^{14} + q^{15} - q^{16} + 2q^{17} - q^{18} + 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 