Properties

 Label 21780x Number of curves $4$ Conductor $21780$ CM no Rank $1$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("x1")

sage: E.isogeny_class()

Elliptic curves in class 21780x

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21780.r4 21780x1 $$[0, 0, 0, 238128, -164337239]$$ $$72268906496/606436875$$ $$-12531100788527310000$$ $$[2]$$ $$276480$$ $$2.3459$$ $$\Gamma_0(N)$$-optimal
21780.r3 21780x2 $$[0, 0, 0, -3437247, -2258565914]$$ $$13584145739344/1195803675$$ $$395351588729596435200$$ $$[2]$$ $$552960$$ $$2.6925$$
21780.r2 21780x3 $$[0, 0, 0, -17011632, -27027819731]$$ $$-26348629355659264/24169921875$$ $$-499434878636718750000$$ $$[2]$$ $$829440$$ $$2.8952$$
21780.r1 21780x4 $$[0, 0, 0, -272246007, -1728981679106]$$ $$6749703004355978704/5671875$$ $$1875211490988000000$$ $$[2]$$ $$1658880$$ $$3.2418$$

Rank

sage: E.rank()

The elliptic curves in class 21780x have rank $$1$$.

Complex multiplication

The elliptic curves in class 21780x do not have complex multiplication.

Modular form 21780.2.a.x

sage: E.q_eigenform(10)

$$q + q^{5} - 2q^{7} - 2q^{13} - 2q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.