# Properties

 Label 600.h Number of curves $6$ Conductor $600$ CM no Rank $0$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 600.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
600.h1 600d5 $$[0, 1, 0, -9608, -365712]$$ $$3065617154/9$$ $$288000000$$ $$$$ $$512$$ $$0.85251$$
600.h2 600d4 $$[0, 1, 0, -1608, 24288]$$ $$28756228/3$$ $$48000000$$ $$$$ $$256$$ $$0.50594$$
600.h3 600d3 $$[0, 1, 0, -608, -5712]$$ $$1556068/81$$ $$1296000000$$ $$[2, 2]$$ $$256$$ $$0.50594$$
600.h4 600d2 $$[0, 1, 0, -108, 288]$$ $$35152/9$$ $$36000000$$ $$[2, 2]$$ $$128$$ $$0.15937$$
600.h5 600d1 $$[0, 1, 0, 17, 38]$$ $$2048/3$$ $$-750000$$ $$$$ $$64$$ $$-0.18721$$ $$\Gamma_0(N)$$-optimal
600.h6 600d6 $$[0, 1, 0, 392, -21712]$$ $$207646/6561$$ $$-209952000000$$ $$$$ $$512$$ $$0.85251$$

## Rank

sage: E.rank()

The elliptic curves in class 600.h have rank $$0$$.

## Complex multiplication

The elliptic curves in class 600.h do not have complex multiplication.

## Modular form600.2.a.h

sage: E.q_eigenform(10)

$$q + q^{3} + q^{9} + 4q^{11} + 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 