# Properties

 Label 51870h Number of curves 4 Conductor 51870 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51870.i1")

sage: E.isogeny_class()

## Elliptic curves in class 51870h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.i3 51870h1 [1, 1, 0, -461968, 120663088] [2] 294912 $$\Gamma_0(N)$$-optimal
51870.i2 51870h2 [1, 1, 0, -461988, 120652092] [2, 2] 589824
51870.i4 51870h3 [1, 1, 0, -415058, 146191398] [2] 1179648
51870.i1 51870h4 [1, 1, 0, -509238, 94409442] [2] 1179648

## Rank

sage: E.rank()

The elliptic curves in class 51870h have rank $$1$$.

## Modular form 51870.2.a.i

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} - q^{13} - q^{14} + q^{15} + q^{16} + 2q^{17} - q^{18} + q^{19} + O(q^{20})$$

## 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.