Properties

Label 10.0.327680000000.2
Degree $10$
Signature $[0, 5]$
Discriminant $-327680000000$
Root discriminant \(14.18\)
Ramified primes $2,5$
Class number $1$
Class group trivial
Galois group $F_5 \wr C_2$ (as 10T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 7*x^8 - 8*x^7 + 18*x^6 - 52*x^5 + 100*x^4 - 120*x^3 + 90*x^2 - 40*x + 10)
 
gp: K = bnfinit(y^10 - 4*y^9 + 7*y^8 - 8*y^7 + 18*y^6 - 52*y^5 + 100*y^4 - 120*y^3 + 90*y^2 - 40*y + 10, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 4*x^9 + 7*x^8 - 8*x^7 + 18*x^6 - 52*x^5 + 100*x^4 - 120*x^3 + 90*x^2 - 40*x + 10);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 4*x^9 + 7*x^8 - 8*x^7 + 18*x^6 - 52*x^5 + 100*x^4 - 120*x^3 + 90*x^2 - 40*x + 10)
 

\( x^{10} - 4x^{9} + 7x^{8} - 8x^{7} + 18x^{6} - 52x^{5} + 100x^{4} - 120x^{3} + 90x^{2} - 40x + 10 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $10$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 5]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-327680000000\) \(\medspace = -\,2^{22}\cdot 5^{7}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(14.18\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Ramified primes:   \(2\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-5}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{963}a^{9}-\frac{236}{963}a^{8}-\frac{44}{321}a^{7}-\frac{200}{963}a^{6}+\frac{194}{963}a^{5}+\frac{67}{321}a^{4}-\frac{308}{963}a^{3}+\frac{74}{963}a^{2}+\frac{256}{963}a+\frac{274}{963}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( \frac{61}{963} a^{9} + \frac{49}{963} a^{8} - \frac{116}{321} a^{7} + \frac{319}{963} a^{6} + \frac{278}{963} a^{5} + \frac{235}{321} a^{4} - \frac{3380}{963} a^{3} + \frac{6440}{963} a^{2} - \frac{5570}{963} a + \frac{2269}{963} \)  (order $4$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{661}{963}a^{9}-\frac{1916}{963}a^{8}+\frac{769}{321}a^{7}-\frac{2195}{963}a^{6}+\frac{8822}{963}a^{5}-\frac{8036}{321}a^{4}+\frac{37162}{963}a^{3}-\frac{31978}{963}a^{2}+\frac{14173}{963}a-\frac{2819}{963}$, $\frac{66}{107}a^{9}-\frac{168}{107}a^{8}+\frac{169}{107}a^{7}-\frac{146}{107}a^{6}+\frac{820}{107}a^{5}-\frac{2142}{107}a^{4}+\frac{2891}{107}a^{3}-\frac{1964}{107}a^{2}+\frac{632}{107}a-\frac{213}{107}$, $\frac{496}{963}a^{9}-\frac{1496}{963}a^{8}+\frac{646}{321}a^{7}-\frac{1937}{963}a^{6}+\frac{6665}{963}a^{5}-\frac{6251}{321}a^{4}+\frac{30202}{963}a^{3}-\frac{27817}{963}a^{2}+\frac{13342}{963}a-\frac{3731}{963}$, $\frac{599}{963}a^{9}-\frac{1729}{963}a^{8}+\frac{608}{321}a^{7}-\frac{1351}{963}a^{6}+\frac{7387}{963}a^{5}-\frac{7054}{321}a^{4}+\frac{30257}{963}a^{3}-\frac{20195}{963}a^{2}+\frac{1190}{963}a+\frac{5231}{963}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 298.181459297 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 298.181459297 \cdot 1}{4\cdot\sqrt{327680000000}}\cr\approx \mathstrut & 1.27524907982 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 7*x^8 - 8*x^7 + 18*x^6 - 52*x^5 + 100*x^4 - 120*x^3 + 90*x^2 - 40*x + 10)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^10 - 4*x^9 + 7*x^8 - 8*x^7 + 18*x^6 - 52*x^5 + 100*x^4 - 120*x^3 + 90*x^2 - 40*x + 10, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^10 - 4*x^9 + 7*x^8 - 8*x^7 + 18*x^6 - 52*x^5 + 100*x^4 - 120*x^3 + 90*x^2 - 40*x + 10);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 4*x^9 + 7*x^8 - 8*x^7 + 18*x^6 - 52*x^5 + 100*x^4 - 120*x^3 + 90*x^2 - 40*x + 10);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$F_5\wr C_2$ (as 10T33):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 800
The 20 conjugacy class representatives for $F_5 \wr C_2$
Character table for $F_5 \wr C_2$

Intermediate fields

\(\Q(\sqrt{-1}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 20 siblings: data not computed
Degree 25 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ R ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.10.0.1}{10} }$ ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.8.20.99$x^{8} + 2 x^{6} + 4 x^{5} + 4 x^{2} + 8 x + 10$$8$$1$$20$$C_4\wr C_2$$[2, 2, 3, 7/2]^{2}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.4.0.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
5.5.7.4$x^{5} + 5 x^{3} + 5$$5$$1$$7$$F_5$$[7/4]_{4}$