Properties

Label 10.0.1828915200000.1
Degree $10$
Signature $[0, 5]$
Discriminant $-1.829\times 10^{12}$
Root discriminant \(16.84\)
Ramified primes $2,3,5,7$
Class number $2$
Class group [2]
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 + 9*x^8 - 16*x^7 + 20*x^6 - 22*x^5 + 26*x^4 - 24*x^3 + 29*x^2 - 14*x + 9)
 
gp: K = bnfinit(y^10 - 4*y^9 + 9*y^8 - 16*y^7 + 20*y^6 - 22*y^5 + 26*y^4 - 24*y^3 + 29*y^2 - 14*y + 9, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 4*x^9 + 9*x^8 - 16*x^7 + 20*x^6 - 22*x^5 + 26*x^4 - 24*x^3 + 29*x^2 - 14*x + 9);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 4*x^9 + 9*x^8 - 16*x^7 + 20*x^6 - 22*x^5 + 26*x^4 - 24*x^3 + 29*x^2 - 14*x + 9)
 

\( x^{10} - 4x^{9} + 9x^{8} - 16x^{7} + 20x^{6} - 22x^{5} + 26x^{4} - 24x^{3} + 29x^{2} - 14x + 9 \) 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:   \(-1828915200000\) \(\medspace = -\,2^{14}\cdot 3^{6}\cdot 5^{5}\cdot 7^{2}\) 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:  \(16.84\)
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:   \(2\), \(3\), \(5\), \(7\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{551}a^{9}+\frac{101}{551}a^{8}+\frac{5}{19}a^{7}-\frac{219}{551}a^{6}+\frac{167}{551}a^{5}-\frac{119}{551}a^{4}+\frac{204}{551}a^{3}-\frac{93}{551}a^{2}+\frac{182}{551}a-\frac{189}{551}$ 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

$C_{2}$, which has order $2$

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:   \( -1 \)  (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:   $\frac{89}{551}a^{9}-\frac{378}{551}a^{8}+\frac{27}{19}a^{7}-\frac{1308}{551}a^{6}+\frac{1639}{551}a^{5}-\frac{1775}{551}a^{4}+\frac{2177}{551}a^{3}-\frac{2216}{551}a^{2}+\frac{1872}{551}a-\frac{842}{551}$, $\frac{117}{551}a^{9}-\frac{305}{551}a^{8}+\frac{15}{19}a^{7}-\frac{828}{551}a^{6}+\frac{805}{551}a^{5}-\frac{699}{551}a^{4}+\frac{1277}{551}a^{3}+\frac{139}{551}a^{2}+\frac{356}{551}a+\frac{478}{551}$, $\frac{61}{551}a^{9}-\frac{451}{551}a^{8}+\frac{39}{19}a^{7}-\frac{1788}{551}a^{6}+\frac{2473}{551}a^{5}-\frac{2300}{551}a^{4}+\frac{2526}{551}a^{3}-\frac{3469}{551}a^{2}+\frac{2286}{551}a-\frac{2162}{551}$, $\frac{22}{551}a^{9}+\frac{18}{551}a^{8}-\frac{4}{19}a^{7}+\frac{141}{551}a^{6}-\frac{183}{551}a^{5}+\frac{137}{551}a^{4}-\frac{471}{551}a^{3}+\frac{709}{551}a^{2}-\frac{404}{551}a+\frac{250}{551}$ 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:  \( 167.222770153 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 167.222770153 \cdot 2}{2\cdot\sqrt{1828915200000}}\cr\approx \mathstrut & 1.21087125807 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 4*x^9 + 9*x^8 - 16*x^7 + 20*x^6 - 22*x^5 + 26*x^4 - 24*x^3 + 29*x^2 - 14*x + 9)
 
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 + 9*x^8 - 16*x^7 + 20*x^6 - 22*x^5 + 26*x^4 - 24*x^3 + 29*x^2 - 14*x + 9, 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 + 9*x^8 - 16*x^7 + 20*x^6 - 22*x^5 + 26*x^4 - 24*x^3 + 29*x^2 - 14*x + 9);
 
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 + 9*x^8 - 16*x^7 + 20*x^6 - 22*x^5 + 26*x^4 - 24*x^3 + 29*x^2 - 14*x + 9);
 
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

\(\Q(\sqrt{-5}) \)

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: 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 R R R ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.10.0.1}{10} }$ ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ ${\href{/padicField/59.10.0.1}{10} }$

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 2.10.14.5$x^{10} + 2 x^{5} + 6$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.5.4.1$x^{5} + 3$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
\(5\) Copy content Toggle raw display 5.10.5.1$x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 4$$1$$1$$0$Trivial$[\ ]$
7.4.2.2$x^{4} - 42 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.0.1$x^{4} + 5 x^{2} + 4 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$