Properties

 Label 112710cv Number of curves 4 Conductor 112710 CM no Rank 0 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("112710.cl1")

sage: E.isogeny_class()

Elliptic curves in class 112710cv

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
112710.cl4 112710cv1 [1, 0, 0, 456614, 20175716] [2] 3317760 $$\Gamma_0(N)$$-optimal
112710.cl3 112710cv2 [1, 0, 0, -1855386, 162132516] [2, 2] 6635520
112710.cl2 112710cv3 [1, 0, 0, -18559586, -30617026404] [2] 13271040
112710.cl1 112710cv4 [1, 0, 0, -22143186, 40035774636] [2] 13271040

Rank

sage: E.rank()

The elliptic curves in class 112710cv have rank $$0$$.

Modular form 112710.2.a.cl

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - 4q^{7} + q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + q^{13} - 4q^{14} - q^{15} + q^{16} + q^{18} + 8q^{19} + O(q^{20})$$

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.