# Properties

 Label 448.e Number of curves $4$ Conductor $448$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 448.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
448.e1 448b3 $$[0, 0, 0, -1196, -15920]$$ $$1443468546/7$$ $$917504$$ $$$$ $$128$$ $$0.34454$$
448.e2 448b4 $$[0, 0, 0, -236, 1104]$$ $$11090466/2401$$ $$314703872$$ $$$$ $$128$$ $$0.34454$$
448.e3 448b2 $$[0, 0, 0, -76, -240]$$ $$740772/49$$ $$3211264$$ $$[2, 2]$$ $$64$$ $$-0.0020328$$
448.e4 448b1 $$[0, 0, 0, 4, -16]$$ $$432/7$$ $$-114688$$ $$$$ $$32$$ $$-0.34861$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 448.e have rank $$1$$.

## Complex multiplication

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

## Modular form448.2.a.e

sage: E.q_eigenform(10)

$$q - 2 q^{5} + q^{7} - 3 q^{9} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 8 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 