# Properties

 Label 1575.h Number of curves $4$ Conductor $1575$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 1575.h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1575.h1 1575f3 $$[1, -1, 0, -25317, 1556716]$$ $$157551496201/13125$$ $$149501953125$$ $$$$ $$3072$$ $$1.1881$$
1575.h2 1575f2 $$[1, -1, 0, -1692, 21091]$$ $$47045881/11025$$ $$125581640625$$ $$[2, 2]$$ $$1536$$ $$0.84149$$
1575.h3 1575f1 $$[1, -1, 0, -567, -4784]$$ $$1771561/105$$ $$1196015625$$ $$$$ $$768$$ $$0.49492$$ $$\Gamma_0(N)$$-optimal
1575.h4 1575f4 $$[1, -1, 0, 3933, 127966]$$ $$590589719/972405$$ $$-11076300703125$$ $$$$ $$3072$$ $$1.1881$$

## Rank

sage: E.rank()

The elliptic curves in class 1575.h have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1575.h do not have complex multiplication.

## Modular form1575.2.a.h

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - q^{7} - 3q^{8} + 6q^{13} - q^{14} - q^{16} + 2q^{17} - 8q^{19} + O(q^{20})$$ ## 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 LMFDB 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 LMFDB labels. 