Properties

 Label 486720.mq Number of curves $4$ Conductor $486720$ CM no Rank $1$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("mq1")

sage: E.isogeny_class()

Elliptic curves in class 486720.mq

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.mq1 486720mq3 $$[0, 0, 0, -1316172, 472688944]$$ $$2186875592/428415$$ $$49397190111029329920$$ $$$$ $$8257536$$ $$2.4956$$ $$\Gamma_0(N)$$-optimal*
486720.mq2 486720mq2 $$[0, 0, 0, -403572, -92027936]$$ $$504358336/38025$$ $$548045748273254400$$ $$[2, 2]$$ $$4128768$$ $$2.1490$$ $$\Gamma_0(N)$$-optimal*
486720.mq3 486720mq1 $$[0, 0, 0, -395967, -95903444]$$ $$30488290624/195$$ $$43913922137280$$ $$$$ $$2064384$$ $$1.8024$$ $$\Gamma_0(N)$$-optimal*
486720.mq4 486720mq4 $$[0, 0, 0, 387348, -408712304]$$ $$55742968/658125$$ $$-75883257453219840000$$ $$$$ $$8257536$$ $$2.4956$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 486720.mq1.

Rank

sage: E.rank()

The elliptic curves in class 486720.mq have rank $$1$$.

Complex multiplication

The elliptic curves in class 486720.mq do not have complex multiplication.

Modular form 486720.2.a.mq

sage: E.q_eigenform(10)

$$q + q^{5} - 2 q^{17} - 4 q^{19} + O(q^{20})$$ 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. 