Properties

 Label 4719.k Number of curves $6$ Conductor $4719$ CM no Rank $1$ Graph

Related objects

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

sage: E.isogeny_class()

Elliptic curves in class 4719.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
4719.k1 4719j3 [1, 0, 1, -830547, -291405149] [2] 30720
4719.k2 4719j5 [1, 0, 1, -364092, 81851779] [2] 61440
4719.k3 4719j4 [1, 0, 1, -57357, -3543245] [2, 2] 30720
4719.k4 4719j2 [1, 0, 1, -51912, -4556015] [2, 2] 15360
4719.k5 4719j1 [1, 0, 1, -2907, -86759] [2] 7680 $$\Gamma_0(N)$$-optimal
4719.k6 4719j6 [1, 0, 1, 162258, -24099209] [2] 61440

Rank

sage: E.rank()

The elliptic curves in class 4719.k have rank $$1$$.

Complex multiplication

The elliptic curves in class 4719.k do not have complex multiplication.

Modular form4719.2.a.k

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - 3q^{8} + q^{9} - 2q^{10} - q^{12} - q^{13} - 2q^{15} - q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

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.