# Properties

 Label 448c Number of curves 6 Conductor 448 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("448.g1")

sage: E.isogeny_class()

## Elliptic curves in class 448c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
448.g5 448c1 [0, -1, 0, -33, 161] [2] 64 $$\Gamma_0(N)$$-optimal
448.g4 448c2 [0, -1, 0, -673, 6945] [2] 128
448.g6 448c3 [0, -1, 0, 287, -3231] [2] 192
448.g3 448c4 [0, -1, 0, -2273, -33439] [2] 384
448.g2 448c5 [0, -1, 0, -10913, -436447] [2] 576
448.g1 448c6 [0, -1, 0, -174753, -28059871] [2] 1152

## Rank

sage: E.rank()

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

## Modular form448.2.a.g

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.