Properties

 Label 435120i Number of curves $6$ Conductor $435120$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("435120.i1")

sage: E.isogeny_class()

Elliptic curves in class 435120i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
435120.i5 435120i1 [0, -1, 0, -662496, 714698496] [2] 15925248 $$\Gamma_0(N)$$-optimal*
435120.i4 435120i2 [0, -1, 0, -16718816, 26263514880] [2, 2] 31850496 $$\Gamma_0(N)$$-optimal*
435120.i1 435120i3 [0, -1, 0, -267347936, 1682621243136] [2] 63700992 $$\Gamma_0(N)$$-optimal*
435120.i3 435120i4 [0, -1, 0, -22990816, 4773134080] [2, 2] 63700992
435120.i6 435120i5 [0, -1, 0, 91316384, 37967944960] [2] 127401984
435120.i2 435120i6 [0, -1, 0, -237650016, -1403906400000] [2] 127401984
*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 435120i1.

Rank

sage: E.rank()

The elliptic curves in class 435120i have rank $$0$$.

Modular form 435120.2.a.i

sage: E.q_eigenform(10)

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

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.