Properties

 Label 4032.k Number of curves 6 Conductor 4032 CM no Rank 0 Graph

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Show commands for: SageMath
sage: E = EllipticCurve("4032.k1")

sage: E.isogeny_class()

Elliptic curves in class 4032.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4032.k1 4032bm5 [0, 0, 0, -451596, -116808176] [2] 16384
4032.k2 4032bm3 [0, 0, 0, -28236, -1823600] [2, 2] 8192
4032.k3 4032bm4 [0, 0, 0, -22476, 1289104] [2] 8192
4032.k4 4032bm6 [0, 0, 0, -19596, -2960624] [2] 16384
4032.k5 4032bm2 [0, 0, 0, -2316, -9200] [2, 2] 4096
4032.k6 4032bm1 [0, 0, 0, 564, -1136] [2] 2048 $$\Gamma_0(N)$$-optimal

Rank

sage: E.rank()

The elliptic curves in class 4032.k have rank $$0$$.

Modular form4032.2.a.k

sage: E.q_eigenform(10)

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.