Properties

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

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("448.a1")

sage: E.isogeny_class()

Elliptic curves in class 448f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
448.a5 448f1 [0, 1, 0, -33, -161] [2] 64 $$\Gamma_0(N)$$-optimal
448.a4 448f2 [0, 1, 0, -673, -6945] [2] 128
448.a6 448f3 [0, 1, 0, 287, 3231] [2] 192
448.a3 448f4 [0, 1, 0, -2273, 33439] [2] 384
448.a2 448f5 [0, 1, 0, -10913, 436447] [2] 576
448.a1 448f6 [0, 1, 0, -174753, 28059871] [2] 1152

Rank

sage: E.rank()

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

Modular form448.2.a.a

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.