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Properties

Label 10.2.3312590600277248.1
Degree $10$
Signature $[2, 4]$
Discriminant $3.313\times 10^{15}$
Root discriminant \(35.65\)
Ramified primes $2,17,1381$
Class number $2$ (GRH)
Class group [2] (GRH)
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 - 3*x^9 + 7*x^8 - 4*x^7 - 17*x^6 + 119*x^5 - 83*x^4 - 311*x^2 + 247*x - 253)
 
gp: K = bnfinit(y^10 - 3*y^9 + 7*y^8 - 4*y^7 - 17*y^6 + 119*y^5 - 83*y^4 - 311*y^2 + 247*y - 253, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - 3*x^9 + 7*x^8 - 4*x^7 - 17*x^6 + 119*x^5 - 83*x^4 - 311*x^2 + 247*x - 253);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - 3*x^9 + 7*x^8 - 4*x^7 - 17*x^6 + 119*x^5 - 83*x^4 - 311*x^2 + 247*x - 253)
 

\( x^{10} - 3x^{9} + 7x^{8} - 4x^{7} - 17x^{6} + 119x^{5} - 83x^{4} - 311x^{2} + 247x - 253 \) 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:   \(3312590600277248\) \(\medspace = 2^{8}\cdot 17^{3}\cdot 1381^{3}\) 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:  \(35.65\)
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:  $2^{4/5}17^{1/2}1381^{1/2}\approx 266.7751031560652$
Ramified primes:   \(2\), \(17\), \(1381\) 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:  \(\Q(\sqrt{23477}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{147025937497}a^{9}+\frac{14763096225}{147025937497}a^{8}-\frac{53574306389}{147025937497}a^{7}+\frac{71882493478}{147025937497}a^{6}-\frac{14130110137}{147025937497}a^{5}-\frac{64976677654}{147025937497}a^{4}+\frac{667179570}{147025937497}a^{3}+\frac{46316326981}{147025937497}a^{2}-\frac{34992820809}{147025937497}a-\frac{38727916516}{147025937497}$ 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 $2$ (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:   $\frac{866722978}{147025937497}a^{9}-\frac{2808230064}{147025937497}a^{8}+\frac{6430901400}{147025937497}a^{7}-\frac{2355700673}{147025937497}a^{6}-\frac{20089457947}{147025937497}a^{5}+\frac{108779009611}{147025937497}a^{4}-\frac{58623603899}{147025937497}a^{3}-\frac{37437134041}{147025937497}a^{2}-\frac{277548513357}{147025937497}a+\frac{287513750631}{147025937497}$, $\frac{3816178636}{147025937497}a^{9}-\frac{14157248635}{147025937497}a^{8}+\frac{30425387081}{147025937497}a^{7}-\frac{23609444501}{147025937497}a^{6}-\frac{75526083878}{147025937497}a^{5}+\frac{499050156374}{147025937497}a^{4}-\frac{555309631892}{147025937497}a^{3}-\frac{252729012466}{147025937497}a^{2}-\frac{1110820275981}{147025937497}a+\frac{2053942040897}{147025937497}$, $\frac{59412094}{147025937497}a^{9}+\frac{414267124}{147025937497}a^{8}+\frac{211671991}{147025937497}a^{7}+\frac{221464484}{147025937497}a^{6}-\frac{875988997}{147025937497}a^{5}+\frac{14766895239}{147025937497}a^{4}+\frac{48526653386}{147025937497}a^{3}+\frac{28931368969}{147025937497}a^{2}-\frac{14203135066}{147025937497}a+\frac{51666244001}{147025937497}$, $\frac{37013048}{147025937497}a^{9}-\frac{436806592}{147025937497}a^{8}+\frac{1765212633}{147025937497}a^{7}-\frac{5041060876}{147025937497}a^{6}+\frac{14729198369}{147025937497}a^{5}-\frac{29854289557}{147025937497}a^{4}+\frac{20012970737}{147025937497}a^{3}-\frac{30019509697}{147025937497}a^{2}+\frac{77129343861}{147025937497}a-\frac{33763449448}{147025937497}$, $\frac{786887080}{147025937497}a^{9}+\frac{274748057}{147025937497}a^{8}+\frac{237689657}{147025937497}a^{7}+\frac{12086928307}{147025937497}a^{6}-\frac{15822641737}{147025937497}a^{5}+\frac{61520980991}{147025937497}a^{4}+\frac{206018355389}{147025937497}a^{3}+\frac{115619214452}{147025937497}a^{2}-\frac{215740970490}{147025937497}a-\frac{437784894567}{147025937497}$ 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:  \( 7013.59815857 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 7013.59815857 \cdot 2}{2\cdot\sqrt{3312590600277248}}\cr\approx \mathstrut & 0.759690090929 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - 3*x^9 + 7*x^8 - 4*x^7 - 17*x^6 + 119*x^5 - 83*x^4 - 311*x^2 + 247*x - 253)
 
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 - 3*x^9 + 7*x^8 - 4*x^7 - 17*x^6 + 119*x^5 - 83*x^4 - 311*x^2 + 247*x - 253, 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 - 3*x^9 + 7*x^8 - 4*x^7 - 17*x^6 + 119*x^5 - 83*x^4 - 311*x^2 + 247*x - 253);
 
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 - 3*x^9 + 7*x^8 - 4*x^7 - 17*x^6 + 119*x^5 - 83*x^4 - 311*x^2 + 247*x - 253);
 
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: 6.2.375632.1, 6.2.207036912517328.1
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: 6.2.375632.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ R ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.5.0.1}{5} }^{2}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 2.10.8.1$x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
\(17\) Copy content Toggle raw display 17.2.1.1$x^{2} + 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.2.2$x^{4} - 272 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(1381\) Copy content Toggle raw display $\Q_{1381}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $3$$1$$3$$0$$C_3$$[\ ]^{3}$
Deg $6$$2$$3$$3$