# Properties

 Label 38640d Number of curves $4$ Conductor $38640$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 38640d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.c3 38640d1 $$[0, -1, 0, -93934356, 350448022656]$$ $$358061097267989271289240144/176126855625$$ $$45088475040000$$ $$[2]$$ $$2211840$$ $$2.8572$$ $$\Gamma_0(N)$$-optimal
38640.c2 38640d2 $$[0, -1, 0, -93934856, 350444105856]$$ $$89516703758060574923008036/1985322833430374025$$ $$2032970581432703001600$$ $$[2, 2]$$ $$4423680$$ $$3.2038$$
38640.c4 38640d3 $$[0, -1, 0, -90581456, 376619404896]$$ $$-40133926989810174413190818/6689384645060302103835$$ $$-13699859753083498708654080$$ $$[2]$$ $$8847360$$ $$3.5503$$
38640.c1 38640d4 $$[0, -1, 0, -97296256, 324018123616]$$ $$49737293673675178002921218/6641736806881023047235$$ $$13602276980492335200737280$$ $$[2]$$ $$8847360$$ $$3.5503$$

## Rank

sage: E.rank()

The elliptic curves in class 38640d have rank $$0$$.

## Complex multiplication

The elliptic curves in class 38640d do not have complex multiplication.

## Modular form 38640.2.a.d

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} - q^{7} + q^{9} - 4q^{11} - 2q^{13} + q^{15} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.