Properties

 Label 441c Number of curves 6 Conductor 441 CM no Rank 1 Graph

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Show commands for: SageMath
sage: E = EllipticCurve("441.f1")

sage: E.isogeny_class()

Elliptic curves in class 441c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
441.f6 441c1 [1, -1, 0, 432, -869] [2] 192 $$\Gamma_0(N)$$-optimal
441.f5 441c2 [1, -1, 0, -1773, -5720] [2, 2] 384
441.f2 441c3 [1, -1, 0, -21618, -1216265] [2, 2] 768
441.f3 441c4 [1, -1, 0, -17208, 867901] [2] 768
441.f1 441c5 [1, -1, 0, -345753, -78165914] [2] 1536
441.f4 441c6 [1, -1, 0, -15003, -1979636] [2] 1536

Rank

sage: E.rank()

The elliptic curves in class 441c have rank $$1$$.

Modular form441.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - q^{16} - 6q^{17} - 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.