Properties

 Label 51870b Number of curves 4 Conductor 51870 CM no Rank 0 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("51870.e1")
sage: E.isogeny_class()

Elliptic curves in class 51870b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.e4 51870b1 [1, 1, 0, -14213, -690483] 2 163840 $$\Gamma_0(N)$$-optimal
51870.e3 51870b2 [1, 1, 0, -230533, -42699827] 4 327680
51870.e2 51870b3 [1, 1, 0, -233653, -41488643] 2 655360
51870.e1 51870b4 [1, 1, 0, -3688533, -2728182627] 2 655360

Rank

sage: E.rank()

The elliptic curves in class 51870b have rank $$0$$.

Modular form None

sage: E.q_eigenform(10)
$$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + q^{10} - q^{12} + q^{13} + q^{14} + q^{15} + q^{16} + 6q^{17} - q^{18} - q^{19} + O(q^{20})$$

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.