Properties

Label 10.2.5100326977536.1
Degree $10$
Signature $[2, 4]$
Discriminant $5.100\times 10^{12}$
Root discriminant \(18.65\)
Ramified primes $2,3,7$
Class number $1$
Class group trivial
Galois group $S_{6}$ (as 10T32)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 2*x^8 - 8*x^7 + 26*x^6 - 20*x^5 + 13*x^4 - 30*x^3 + 23*x^2 + 20*x - 19)
 
gp: K = bnfinit(y^10 - 2*y^9 + 2*y^8 - 8*y^7 + 26*y^6 - 20*y^5 + 13*y^4 - 30*y^3 + 23*y^2 + 20*y - 19, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 2*x^9 + 2*x^8 - 8*x^7 + 26*x^6 - 20*x^5 + 13*x^4 - 30*x^3 + 23*x^2 + 20*x - 19);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 2*x^9 + 2*x^8 - 8*x^7 + 26*x^6 - 20*x^5 + 13*x^4 - 30*x^3 + 23*x^2 + 20*x - 19)
 

\( x^{10} - 2x^{9} + 2x^{8} - 8x^{7} + 26x^{6} - 20x^{5} + 13x^{4} - 30x^{3} + 23x^{2} + 20x - 19 \) 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:   \(5100326977536\) \(\medspace = 2^{15}\cdot 3^{3}\cdot 7^{8}\) 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:  \(18.65\)
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))
 
Galois root discriminant:  $2^{9/4}3^{1/2}7^{4/5}\approx 39.080179481705656$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q(\sqrt{6}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{505317}a^{9}+\frac{27880}{168439}a^{8}-\frac{42848}{168439}a^{7}+\frac{120992}{505317}a^{6}+\frac{197770}{505317}a^{5}-\frac{59664}{168439}a^{4}-\frac{53653}{505317}a^{3}+\frac{244460}{505317}a^{2}-\frac{7915}{168439}a+\frac{153418}{505317}$ 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:  $5$
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:   $\frac{12836}{168439}a^{9}-\frac{27146}{168439}a^{8}+\frac{37660}{168439}a^{7}-\frac{122707}{168439}a^{6}+\frac{368429}{168439}a^{5}-\frac{370230}{168439}a^{4}+\frac{394041}{168439}a^{3}-\frac{635327}{168439}a^{2}+\frac{589087}{168439}a-\frac{115340}{168439}$, $\frac{26363}{505317}a^{9}-\frac{33629}{505317}a^{8}+\frac{18769}{505317}a^{7}-\frac{185686}{505317}a^{6}+\frac{455021}{505317}a^{5}+\frac{52489}{505317}a^{4}+\frac{96683}{505317}a^{3}-\frac{619355}{505317}a^{2}-\frac{575428}{505317}a+\frac{338344}{505317}$, $\frac{101123}{505317}a^{9}-\frac{22742}{168439}a^{8}+\frac{6532}{168439}a^{7}-\frac{671822}{505317}a^{6}+\frac{1680752}{505317}a^{5}+\frac{250747}{168439}a^{4}+\frac{541627}{505317}a^{3}-\frac{2105645}{505317}a^{2}-\frac{134856}{168439}a+\frac{2372465}{505317}$, $\frac{6130}{505317}a^{9}-\frac{15116}{505317}a^{8}-\frac{17078}{505317}a^{7}+\frac{14681}{168439}a^{6}+\frac{74617}{505317}a^{5}-\frac{9314}{505317}a^{4}-\frac{257906}{168439}a^{3}+\frac{780212}{505317}a^{2}+\frac{311324}{505317}a-\frac{205451}{168439}$, $\frac{20519}{505317}a^{9}-\frac{15811}{505317}a^{8}-\frac{8035}{505317}a^{7}-\frac{52004}{168439}a^{6}+\frac{347120}{505317}a^{5}+\frac{243986}{505317}a^{4}+\frac{3799}{168439}a^{3}-\frac{712436}{505317}a^{2}-\frac{940262}{505317}a+\frac{346035}{168439}$ 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:  \( 603.818540065 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 603.818540065 \cdot 1}{2\cdot\sqrt{5100326977536}}\cr\approx \mathstrut & 0.833406538223 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 2*x^9 + 2*x^8 - 8*x^7 + 26*x^6 - 20*x^5 + 13*x^4 - 30*x^3 + 23*x^2 + 20*x - 19)
 
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 - 2*x^9 + 2*x^8 - 8*x^7 + 26*x^6 - 20*x^5 + 13*x^4 - 30*x^3 + 23*x^2 + 20*x - 19, 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 - 2*x^9 + 2*x^8 - 8*x^7 + 26*x^6 - 20*x^5 + 13*x^4 - 30*x^3 + 23*x^2 + 20*x - 19);
 
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 - 2*x^9 + 2*x^8 - 8*x^7 + 26*x^6 - 20*x^5 + 13*x^4 - 30*x^3 + 23*x^2 + 20*x - 19);
 
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

$S_6$ (as 10T32):

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 non-solvable group of order 720
The 11 conjugacy class representatives for $S_{6}$
Character table for $S_{6}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 6 siblings: 6.2.3687936.1, 6.2.33191424.1
Degree 12 siblings: data not computed
Degree 15 siblings: data not computed
Degree 20 siblings: data not computed
Degree 30 siblings: data not computed
Degree 36 sibling: data not computed
Degree 40 siblings: data not computed
Degree 45 sibling: data not computed
Minimal sibling: 6.2.3687936.1

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 ${\href{/padicField/5.5.0.1}{5} }^{2}$ R ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.3.0.1}{3} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.5.0.1}{5} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.6.9.8$x^{6} - 8 x^{5} + 62 x^{4} + 224 x^{3} + 316 x^{2} + 1184 x + 1608$$2$$3$$9$$A_4\times C_2$$[2, 2, 3]^{3}$
\(3\) Copy content Toggle raw display 3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.0.1$x^{4} + 2 x^{3} + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(7\) Copy content Toggle raw display 7.10.8.1$x^{10} + 30 x^{9} + 375 x^{8} + 2520 x^{7} + 9810 x^{6} + 22370 x^{5} + 29640 x^{4} + 24780 x^{3} + 21465 x^{2} + 33300 x + 33934$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$