Properties

 Label 129360.w Number of curves $6$ Conductor $129360$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("129360.w1")

sage: E.isogeny_class()

Elliptic curves in class 129360.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
129360.w1 129360du6 [0, -1, 0, -10388016, -12882777984] [2] 6291456
129360.w2 129360du4 [0, -1, 0, -686016, -177038784] [2, 2] 3145728
129360.w3 129360du2 [0, -1, 0, -211696, 35077120] [2, 2] 1572864
129360.w4 129360du1 [0, -1, 0, -207776, 36522816] [2] 786432 $$\Gamma_0(N)$$-optimal
129360.w5 129360du3 [0, -1, 0, 199904, 154605760] [2] 3145728
129360.w6 129360du5 [0, -1, 0, 1426864, -1054306560] [2] 6291456

Rank

sage: E.rank()

The elliptic curves in class 129360.w have rank $$1$$.

Modular form 129360.2.a.w

sage: E.q_eigenform(10)

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.