# Properties

 Label 10830h Number of curves 4 Conductor 10830 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("10830.h1")

sage: E.isogeny_class()

## Elliptic curves in class 10830h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10830.h4 10830h1 [1, 1, 0, -3617, -144411]  34560 $$\Gamma_0(N)$$-optimal
10830.h3 10830h2 [1, 1, 0, -68597, -6941319] [2, 2] 69120
10830.h1 10830h3 [1, 1, 0, -1097447, -442967949]  138240
10830.h2 10830h4 [1, 1, 0, -79427, -4617201]  138240

## Rank

sage: E.rank()

The elliptic curves in class 10830h have rank $$1$$.

## Modular form 10830.2.a.h

sage: E.q_eigenform(10)

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

## 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. 