# Properties

 Label 9025.f Number of curves $2$ Conductor $9025$ CM $$\Q(\sqrt{-19})$$ Rank $1$ Graph

# Related objects

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

E.isogeny_class()

## Elliptic curves in class 9025.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
9025.f1 9025a2 $$[0, 0, 1, -342950, -77378094]$$ $$-884736$$ $$-5041995277796875$$ $$[]$$ $$53200$$ $$1.9268$$   $$-19$$
9025.f2 9025a1 $$[0, 0, 1, -950, 11281]$$ $$-884736$$ $$-107171875$$ $$[]$$ $$2800$$ $$0.45457$$ $$\Gamma_0(N)$$-optimal $$-19$$

## Rank

sage: E.rank()

The elliptic curves in class 9025.f have rank $$1$$.

## Complex multiplication

Each elliptic curve in class 9025.f has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-19})$$.

## Modular form9025.2.a.f

sage: E.q_eigenform(10)

$$q - 2 q^{4} - 3 q^{7} - 3 q^{9} - 5 q^{11} + 4 q^{16} + 7 q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 19 \\ 19 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.