Properties

 Label 479370t Number of curves $4$ Conductor $479370$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("479370.t1")

sage: E.isogeny_class()

Elliptic curves in class 479370t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
479370.t4 479370t1 [1, 1, 0, -8427, 504621] [2] 2322432 $$\Gamma_0(N)$$-optimal*
479370.t3 479370t2 [1, 1, 0, -159807, 24513489] [2, 2] 4644864 $$\Gamma_0(N)$$-optimal*
479370.t1 479370t3 [1, 1, 0, -2556657, 1572399219] [2] 9289728 $$\Gamma_0(N)$$-optimal*
479370.t2 479370t4 [1, 1, 0, -185037, 16222911] [2] 9289728
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 479370t1.

Rank

sage: E.rank()

The elliptic curves in class 479370t have rank $$1$$.

Modular form 479370.2.a.t

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 4q^{7} - q^{8} + q^{9} - q^{10} + 4q^{11} - q^{12} - 2q^{13} - 4q^{14} - q^{15} + q^{16} + 2q^{17} - q^{18} + q^{19} + O(q^{20})$$

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.