# Properties

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

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 448a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
448.d4 448a1 [0, 0, 0, 4, 16] [2] 32 $$\Gamma_0(N)$$-optimal
448.d3 448a2 [0, 0, 0, -76, 240] [2, 2] 64
448.d2 448a3 [0, 0, 0, -236, -1104] [2] 128
448.d1 448a4 [0, 0, 0, -1196, 15920] [4] 128

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form448.2.a.a

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.