# Properties

 Label 1575.f Number of curves $3$ Conductor $1575$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 1575.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1575.f1 1575e3 $$[0, 0, 1, -29550, 2045281]$$ $$-250523582464/13671875$$ $$-155731201171875$$ $$[]$$ $$4320$$ $$1.4815$$
1575.f2 1575e1 $$[0, 0, 1, -300, -2219]$$ $$-262144/35$$ $$-398671875$$ $$[]$$ $$480$$ $$0.38287$$ $$\Gamma_0(N)$$-optimal
1575.f3 1575e2 $$[0, 0, 1, 1950, 5656]$$ $$71991296/42875$$ $$-488373046875$$ $$[]$$ $$1440$$ $$0.93218$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1575.2.a.f

sage: E.q_eigenform(10)

$$q - 2q^{4} - q^{7} + 3q^{11} - 5q^{13} + 4q^{16} + 3q^{17} + 2q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 