Properties

Label 10.0.1533949635.1
Degree 1010
Signature [0,5][0, 5]
Discriminant 1533949635-1533949635
Root discriminant 8.298.29
Ramified primes 3,5,29,269,131093,5,29,269,13109
Class number 11
Class group trivial
Galois group S10S_{10} (as 10T45)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

x10+2x8x7+3x63x5+4x45x3+3x22x+1 x^{10} + 2x^{8} - x^{7} + 3x^{6} - 3x^{5} + 4x^{4} - 5x^{3} + 3x^{2} - 2x + 1 Copy content Toggle raw display

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

Invariants

Degree:  1010
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  [0,5][0, 5]
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   1533949635-1533949635 =352926913109\medspace = -\,3\cdot 5\cdot 29\cdot 269\cdot 13109 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:  8.298.29
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:  31/251/2291/22691/2131091/239165.669086586533^{1/2}5^{1/2}29^{1/2}269^{1/2}13109^{1/2}\approx 39165.66908658653
Ramified primes:   33, 55, 2929, 269269, 1310913109 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(1533949635\Q(\sqrt{-1533949635})
Aut(K/Q)\Aut(K/\Q):   C1C_1
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over Q\Q.
This is not a CM field.

Integral basis (with respect to field generator aa)

11, aa, a2a^{2}, a3a^{3}, a4a^{4}, a5a^{5}, a6a^{6}, a7a^{7}, a8a^{8}, 15a915a825a7+15a6+25a515a3+15a2+25a+15\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{1}{5} Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  11
Inessential primes:  None

Class group and class number

Trivial group, which has order 11

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:  44
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   1 -1  (order 22) 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:   aa, 15a915a825a745a635a5a415a3+15a235a+65\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{4}{5}a^{6}-\frac{3}{5}a^{5}-a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{3}{5}a+\frac{6}{5}, 25a9+35a8+15a7+25a615a575a3+25a2115a+75\frac{2}{5}a^{9}+\frac{3}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{7}{5}a^{3}+\frac{2}{5}a^{2}-\frac{11}{5}a+\frac{7}{5}, 35a9+25a8+95a7+35a6+115a5a4+125a3125a2+15a75\frac{3}{5}a^{9}+\frac{2}{5}a^{8}+\frac{9}{5}a^{7}+\frac{3}{5}a^{6}+\frac{11}{5}a^{5}-a^{4}+\frac{12}{5}a^{3}-\frac{12}{5}a^{2}+\frac{1}{5}a-\frac{7}{5} 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:  2.619339148587336 2.619339148587336
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

lims1(s1)ζK(s)=(2r1(2π)r2RhwD(20(2π)52.619339148587336121533949635(0.327457943363349 \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 2.619339148587336 \cdot 1}{2\cdot\sqrt{1533949635}}\cr\approx \mathstrut & 0.327457943363349 \end{aligned}

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 + 2*x^8 - x^7 + 3*x^6 - 3*x^5 + 4*x^4 - 5*x^3 + 3*x^2 - 2*x + 1)
 
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^8 - x^7 + 3*x^6 - 3*x^5 + 4*x^4 - 5*x^3 + 3*x^2 - 2*x + 1, 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(Rationals()); K<a> := NumberField(x^10 + 2*x^8 - x^7 + 3*x^6 - 3*x^5 + 4*x^4 - 5*x^3 + 3*x^2 - 2*x + 1);
 
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^8 - x^7 + 3*x^6 - 3*x^5 + 4*x^4 - 5*x^3 + 3*x^2 - 2*x + 1);
 
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

S10S_{10} (as 10T45):

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 3628800
The 42 conjugacy class representatives for S10S_{10}
Character table for S10S_{10}

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and Q\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 20 sibling: data not computed
Degree 45 sibling: data not computed
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

pp 22 33 55 77 1111 1313 1717 1919 2323 2929 3131 3737 4141 4343 4747 5353 5959
Cycle type 10{\href{/padicField/2.10.0.1}{10} } R R 5,3,2{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} } 5,3,2{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} } 8,2{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} } 8,2{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} } 7,2,1{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} } 5,3,2{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} } R 42,2{\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} } 4,3,2,1{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} } 10{\href{/padicField/41.10.0.1}{10} } 7,2,1{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} } 5,3,12{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2} 5,2,13{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3} 6,3,1{\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=7 in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\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 [ei,fi][e_i,f_i] for the factorization of the ideal pOKp\mathcal{O}_K for p=7p=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

ppLabelPolynomial ee ff cc Galois group Slope content
33 Copy content Toggle raw display 3.1.2.1a1.2x2+6x^{2} + 6221111C2C_2[ ]2[\ ]_{2}
3.8.1.0a1.1x8+2x5+x4+2x2+2x+2x^{8} + 2 x^{5} + x^{4} + 2 x^{2} + 2 x + 2118800C8C_8[ ]8[\ ]^{8}
55 Copy content Toggle raw display 5.1.2.1a1.1x2+5x^{2} + 5221111C2C_2[ ]2[\ ]_{2}
5.8.1.0a1.1x8+x4+3x2+4x+2x^{8} + x^{4} + 3 x^{2} + 4 x + 2118800C8C_8[ ]8[\ ]^{8}
2929 Copy content Toggle raw display 29.1.2.1a1.2x2+58x^{2} + 58221111C2C_2[ ]2[\ ]_{2}
29.8.1.0a1.1x8+3x4+24x3+26x2+23x+2x^{8} + 3 x^{4} + 24 x^{3} + 26 x^{2} + 23 x + 2118800C8C_8[ ]8[\ ]^{8}
269269 Copy content Toggle raw display Q269\Q_{269}xx111100Trivial[ ][\ ]
Q269\Q_{269}xx111100Trivial[ ][\ ]
Deg 22221111C2C_2[ ]2[\ ]_{2}
Deg 33113300C3C_3[ ]3[\ ]^{3}
Deg 33113300C3C_3[ ]3[\ ]^{3}
1310913109 Copy content Toggle raw display Q13109\Q_{13109}xx111100Trivial[ ][\ ]
Deg 22112200C2C_2[ ]2[\ ]^{2}
Deg 22221111C2C_2[ ]2[\ ]_{2}
Deg 55115500C5C_5[ ]5[\ ]^{5}

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)