# Properties

 Label 448.f Number of curves $2$ Conductor $448$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 448.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
448.f1 448d2 $$[0, -1, 0, -33, -31]$$ $$125000/49$$ $$1605632$$ $$$$ $$64$$ $$-0.11063$$
448.f2 448d1 $$[0, -1, 0, 7, -7]$$ $$8000/7$$ $$-28672$$ $$$$ $$32$$ $$-0.45720$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 448.f have rank $$0$$.

## Complex multiplication

The elliptic curves in class 448.f do not have complex multiplication.

## Modular form448.2.a.f

sage: E.q_eigenform(10)

$$q + 2q^{3} - q^{7} + q^{9} + 4q^{11} + 4q^{13} - 2q^{17} + 6q^{19} + O(q^{20})$$ ## 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. 