# Properties

 Label 9075.a Number of curves $2$ Conductor $9075$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 9075.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
9075.a1 9075j1 $$[0, -1, 1, -1008, 12968]$$ $$-102400/3$$ $$-3321676875$$ $$[]$$ $$8400$$ $$0.60585$$ $$\Gamma_0(N)$$-optimal
9075.a2 9075j2 $$[0, -1, 1, 5042, -610182]$$ $$20480/243$$ $$-168159891796875$$ $$[]$$ $$42000$$ $$1.4106$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form9075.2.a.a

sage: E.q_eigenform(10)

$$q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{6} + 3 q^{7} + q^{9} - 2 q^{12} - q^{13} - 6 q^{14} - 4 q^{16} - 2 q^{17} - 2 q^{18} + 5 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.