# Properties

 Label 448.d Number of curves $4$ Conductor $448$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
E = EllipticCurve("d1")

E.isogeny_class()

## Elliptic curves in class 448.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
448.d1 448a4 $$[0, 0, 0, -1196, 15920]$$ $$1443468546/7$$ $$917504$$ $$[4]$$ $$128$$ $$0.34454$$
448.d2 448a3 $$[0, 0, 0, -236, -1104]$$ $$11090466/2401$$ $$314703872$$ $$[2]$$ $$128$$ $$0.34454$$
448.d3 448a2 $$[0, 0, 0, -76, 240]$$ $$740772/49$$ $$3211264$$ $$[2, 2]$$ $$64$$ $$-0.0020328$$
448.d4 448a1 $$[0, 0, 0, 4, 16]$$ $$432/7$$ $$-114688$$ $$[2]$$ $$32$$ $$-0.34861$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form448.2.a.d

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.