Properties

 Label 105a Number of curves 4 Conductor 105 CM no Rank 0 Graph Related objects

Show commands for: SageMath
sage: E = EllipticCurve("105.a1")
sage: E.isogeny_class()

Elliptic curves in class 105a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
105.a3 105a1 [1, 0, 1, -3, 1] 2 4 $$\Gamma_0(N)$$-optimal
105.a2 105a2 [1, 0, 1, -8, -7] 4 8
105.a1 105a3 [1, 0, 1, -113, -469] 2 16
105.a4 105a4 [1, 0, 1, 17, -37] 4 16

Rank

sage: E.rank()

The elliptic curves in class 105a have rank $$0$$.

Modular form105.2.a.a

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

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. 