# Properties

 Label 9075.f Number of curves $2$ Conductor $9075$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 9075.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9075.f1 9075s2 $$[1, 0, 0, -856138, -303556483]$$ $$15069223/81$$ $$373034693302734375$$ $$$$ $$168960$$ $$2.2158$$
9075.f2 9075s1 $$[1, 0, 0, -24263, -9904608]$$ $$-343/9$$ $$-41448299255859375$$ $$$$ $$84480$$ $$1.8692$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 9075.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 9075.f do not have complex multiplication.

## Modular form9075.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{6} - 4 q^{7} + 3 q^{8} + q^{9} - q^{12} - 2 q^{13} + 4 q^{14} - q^{16} - 2 q^{17} - q^{18} + 8 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 