Properties

Label 10.2.127401984000000.5
Degree $10$
Signature $[2, 4]$
Discriminant $1.274\times 10^{14}$
Root discriminant \(25.73\)
Ramified primes $2,3,5$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $(D_5 \wr C_2):C_2$ (as 10T27)

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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 + 16*x^7 - 11*x^6 - 32*x^5 - 12*x^4 + 20*x^3 + 53*x^2 + 32*x + 10)
 
gp: K = bnfinit(y^10 - 4*y^9 + 16*y^7 - 11*y^6 - 32*y^5 - 12*y^4 + 20*y^3 + 53*y^2 + 32*y + 10, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 4*x^9 + 16*x^7 - 11*x^6 - 32*x^5 - 12*x^4 + 20*x^3 + 53*x^2 + 32*x + 10);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 4*x^9 + 16*x^7 - 11*x^6 - 32*x^5 - 12*x^4 + 20*x^3 + 53*x^2 + 32*x + 10)
 

\( x^{10} - 4x^{9} + 16x^{7} - 11x^{6} - 32x^{5} - 12x^{4} + 20x^{3} + 53x^{2} + 32x + 10 \) 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:  $[2, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(127401984000000\) \(\medspace = 2^{25}\cdot 3^{5}\cdot 5^{6}\) 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:  \(25.73\)
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:   \(2\), \(3\), \(5\) 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:  \(\Q(\sqrt{6}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{7}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a$, $\frac{1}{5}a^{8}+\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{45}a^{9}+\frac{1}{45}a^{8}-\frac{4}{45}a^{7}-\frac{4}{45}a^{6}+\frac{1}{9}a^{5}-\frac{7}{45}a^{4}-\frac{2}{45}a^{3}-\frac{17}{45}a^{2}+\frac{13}{45}a-\frac{4}{9}$ 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 $1$ (assuming GRH)

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:  $5$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{1}{45}a^{9}-\frac{17}{45}a^{8}+\frac{68}{45}a^{7}-\frac{67}{45}a^{6}-\frac{121}{45}a^{5}+\frac{236}{45}a^{4}+\frac{23}{9}a^{3}+\frac{1}{45}a^{2}-\frac{302}{45}a-\frac{49}{9}$, $\frac{2}{5}a^{9}-\frac{12}{5}a^{8}+\frac{23}{5}a^{7}-\frac{6}{5}a^{6}-\frac{34}{5}a^{5}+\frac{28}{5}a^{4}-\frac{47}{5}a^{3}+\frac{47}{5}a^{2}+\frac{43}{5}a+5$, $\frac{1}{9}a^{9}-\frac{22}{45}a^{8}+\frac{7}{45}a^{7}+\frac{88}{45}a^{6}-\frac{101}{45}a^{5}-\frac{152}{45}a^{4}+\frac{7}{9}a^{3}+\frac{113}{45}a^{2}+\frac{263}{45}a+\frac{25}{9}$, $\frac{2}{5}a^{9}-\frac{11}{5}a^{8}+\frac{16}{5}a^{7}+2a^{6}-\frac{38}{5}a^{5}-2a^{4}-a^{3}+\frac{54}{5}a^{2}+\frac{37}{5}a+3$, $\frac{28}{15}a^{9}-\frac{149}{15}a^{8}+\frac{197}{15}a^{7}+\frac{37}{3}a^{6}-\frac{541}{15}a^{5}-\frac{193}{15}a^{4}-\frac{101}{15}a^{3}+\frac{733}{15}a^{2}+\frac{514}{15}a+\frac{41}{3}$ Copy content Toggle raw display (assuming GRH)
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:  \( 6183.05357883 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 6183.05357883 \cdot 1}{2\cdot\sqrt{127401984000000}}\cr\approx \mathstrut & 1.70751443320 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 16*x^7 - 11*x^6 - 32*x^5 - 12*x^4 + 20*x^3 + 53*x^2 + 32*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 + 16*x^7 - 11*x^6 - 32*x^5 - 12*x^4 + 20*x^3 + 53*x^2 + 32*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 + 16*x^7 - 11*x^6 - 32*x^5 - 12*x^4 + 20*x^3 + 53*x^2 + 32*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 + 16*x^7 - 11*x^6 - 32*x^5 - 12*x^4 + 20*x^3 + 53*x^2 + 32*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

$D_5^2.C_2^2$ (as 10T27):

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 400
The 16 conjugacy class representatives for $(D_5 \wr C_2):C_2$
Character table for $(D_5 \wr C_2):C_2$

Intermediate fields

\(\Q(\sqrt{6}) \)

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 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 10.2.23592960000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R R ${\href{/padicField/7.10.0.1}{10} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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 2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.8.22.20$x^{8} + 16 x^{7} + 60 x^{6} - 40 x^{5} + 48 x^{4} + 160 x^{3} + 424 x^{2} + 240 x + 300$$4$$2$$22$$Q_8:C_2$$[2, 3, 4]^{2}$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.8.4.1$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.4.3.2$x^{4} + 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} + 20$$4$$1$$3$$C_4$$[\ ]_{4}$