Properties

 Label 162240.w Number of curves $4$ Conductor $162240$ CM no Rank $1$ Graph Related objects

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

sage: E.isogeny_class()

Elliptic curves in class 162240.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
162240.w1 162240hx3 $$[0, -1, 0, -146241, 17555745]$$ $$2186875592/428415$$ $$67760205913620480$$ $$$$ $$1032192$$ $$1.9463$$
162240.w2 162240hx2 $$[0, -1, 0, -44841, -3393495]$$ $$504358336/38025$$ $$751777432473600$$ $$[2, 2]$$ $$516096$$ $$1.5997$$
162240.w3 162240hx1 $$[0, -1, 0, -43996, -3537314]$$ $$30488290624/195$$ $$60238576320$$ $$$$ $$258048$$ $$1.2531$$ $$\Gamma_0(N)$$-optimal
162240.w4 162240hx4 $$[0, -1, 0, 43039, -15151839]$$ $$55742968/658125$$ $$-104092259880960000$$ $$$$ $$1032192$$ $$1.9463$$

Rank

sage: E.rank()

The elliptic curves in class 162240.w have rank $$1$$.

Complex multiplication

The elliptic curves in class 162240.w do not have complex multiplication.

Modular form 162240.2.a.w

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} + q^{9} + q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})$$ 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 LMFDB 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 LMFDB labels. 