# Properties

 Label 250173h Number of curves 4 Conductor 250173 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("250173.h1")

sage: E.isogeny_class()

## Elliptic curves in class 250173h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
250173.h4 250173h1 [1, -1, 1, -250241, -154311024]  4976640 $$\Gamma_0(N)$$-optimal
250173.h3 250173h2 [1, -1, 1, -6114686, -5807636004] [2, 2] 9953280
250173.h2 250173h3 [1, -1, 1, -8275271, -1338681990]  19906560
250173.h1 250173h4 [1, -1, 1, -97785221, -372159762078]  19906560

## Rank

sage: E.rank()

The elliptic curves in class 250173h have rank $$1$$.

## Modular form 250173.2.a.h

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 