Properties

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

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("4032.e1")

sage: E.isogeny_class()

Elliptic curves in class 4032.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4032.e1 4032i3 [0, 0, 0, -774156, -262174736] [2] 24576
4032.e2 4032i5 [0, 0, 0, -526476, 145621744] [2] 49152
4032.e3 4032i4 [0, 0, 0, -59916, -1997840] [2, 2] 24576
4032.e4 4032i2 [0, 0, 0, -48396, -4094480] [2, 2] 12288
4032.e5 4032i1 [0, 0, 0, -2316, -94736] [2] 6144 $$\Gamma_0(N)$$-optimal
4032.e6 4032i6 [0, 0, 0, 222324, -15432464] [2] 49152

Rank

sage: E.rank()

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

Modular form4032.2.a.e

sage: E.q_eigenform(10)

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.