# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 1575f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1575.h3 1575f1 [1, -1, 0, -567, -4784]  768 $$\Gamma_0(N)$$-optimal
1575.h2 1575f2 [1, -1, 0, -1692, 21091] [2, 2] 1536
1575.h1 1575f3 [1, -1, 0, -25317, 1556716]  3072
1575.h4 1575f4 [1, -1, 0, 3933, 127966]  3072

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1575.2.a.f

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 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. 