Properties

 Label 1470p Number of curves $6$ Conductor $1470$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("1470.q1")

sage: E.isogeny_class()

Elliptic curves in class 1470p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1470.q6 1470p1 [1, 0, 0, 489, 5865] [2] 1536 $$\Gamma_0(N)$$-optimal
1470.q5 1470p2 [1, 0, 0, -3431, 59961] [2, 2] 3072
1470.q4 1470p3 [1, 0, 0, -18131, -889659] [2] 6144
1470.q2 1470p4 [1, 0, 0, -51451, 4487405] [2, 2] 6144
1470.q1 1470p5 [1, 0, 0, -823201, 287410955] [2] 12288
1470.q3 1470p6 [1, 0, 0, -48021, 5112351] [2] 12288

Rank

sage: E.rank()

The elliptic curves in class 1470p have rank $$0$$.

Modular form1470.2.a.q

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 2q^{13} - q^{15} + q^{16} - 2q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.