# Properties

 Label 9075.i Number of curves $2$ Conductor $9075$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("i1")

sage: E.isogeny_class()

## Elliptic curves in class 9075.i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9075.i1 9075f1 $$[0, -1, 1, -2823, -59632]$$ $$-56197120/3267$$ $$-144692244675$$ $$[]$$ $$8640$$ $$0.89774$$ $$\Gamma_0(N)$$-optimal
9075.i2 9075f2 $$[0, -1, 1, 15327, -112267]$$ $$8990228480/5314683$$ $$-235382128254075$$ $$[]$$ $$25920$$ $$1.4470$$

## Rank

sage: E.rank()

The elliptic curves in class 9075.i have rank $$0$$.

## Complex multiplication

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

## Modular form9075.2.a.i

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 