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Properties

Label 10.0.46584877058339283.1
Degree $10$
Signature $[0, 5]$
Discriminant $-4.658\times 10^{16}$
Root discriminant \(46.43\)
Ramified primes $3,61$
Class number $25$
Class group [5, 5]
Galois group $C_{10}$ (as 10T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 25*x^8 - 10*x^7 + 552*x^6 - 313*x^5 + 1286*x^4 + 1321*x^3 + 1460*x^2 + 533*x + 169)
 
gp: K = bnfinit(y^10 - y^9 + 25*y^8 - 10*y^7 + 552*y^6 - 313*y^5 + 1286*y^4 + 1321*y^3 + 1460*y^2 + 533*y + 169, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 + 25*x^8 - 10*x^7 + 552*x^6 - 313*x^5 + 1286*x^4 + 1321*x^3 + 1460*x^2 + 533*x + 169);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^10 - x^9 + 25*x^8 - 10*x^7 + 552*x^6 - 313*x^5 + 1286*x^4 + 1321*x^3 + 1460*x^2 + 533*x + 169)
 

\( x^{10} - x^{9} + 25x^{8} - 10x^{7} + 552x^{6} - 313x^{5} + 1286x^{4} + 1321x^{3} + 1460x^{2} + 533x + 169 \) 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:   \(-46584877058339283\) \(\medspace = -\,3^{5}\cdot 61^{8}\) 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:  \(46.43\)
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:  $3^{1/2}61^{4/5}\approx 46.43275958595107$
Ramified primes:   \(3\), \(61\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:  $10$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(183=3\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{183}(1,·)$, $\chi_{183}(34,·)$, $\chi_{183}(131,·)$, $\chi_{183}(70,·)$, $\chi_{183}(142,·)$, $\chi_{183}(20,·)$, $\chi_{183}(119,·)$, $\chi_{183}(58,·)$, $\chi_{183}(62,·)$, $\chi_{183}(95,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-3}) \), 10.0.46584877058339283.1$^{15}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{377}a^{8}+\frac{162}{377}a^{7}-\frac{167}{377}a^{6}+\frac{147}{377}a^{5}+\frac{60}{377}a^{4}+\frac{3}{377}a^{3}-\frac{149}{377}a^{2}+\frac{161}{377}a-\frac{9}{29}$, $\frac{1}{772048339283}a^{9}-\frac{420613759}{772048339283}a^{8}-\frac{8001464183}{772048339283}a^{7}+\frac{100047512725}{772048339283}a^{6}+\frac{159216746515}{772048339283}a^{5}-\frac{124248767497}{772048339283}a^{4}+\frac{90668903475}{772048339283}a^{3}+\frac{142973400302}{772048339283}a^{2}+\frac{985664971}{2047873579}a+\frac{29026943053}{59388333791}$ 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

$C_{5}\times C_{5}$, which has order $25$

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:   \( -\frac{51390875}{26622356527} a^{9} + \frac{54609062}{26622356527} a^{8} - \frac{1298391493}{26622356527} a^{7} + \frac{48183693}{2047873579} a^{6} - \frac{2204955674}{2047873579} a^{5} + \frac{1420115198}{2047873579} a^{4} - \frac{72683143393}{26622356527} a^{3} - \frac{48242785109}{26622356527} a^{2} - \frac{83093960998}{26622356527} a - \frac{287163487}{2047873579} \)  (order $6$) 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:   $\frac{3744338516}{772048339283}a^{9}-\frac{3341830827}{772048339283}a^{8}+\frac{92177547278}{772048339283}a^{7}-\frac{25625995799}{772048339283}a^{6}+\frac{2033670209166}{772048339283}a^{5}-\frac{31804569703}{26622356527}a^{4}+\frac{4122384789979}{772048339283}a^{3}+\frac{498223727480}{59388333791}a^{2}+\frac{4619582843218}{772048339283}a+\frac{129562377967}{59388333791}$, $\frac{2281158160}{772048339283}a^{9}-\frac{249428670}{26622356527}a^{8}+\frac{59738166465}{772048339283}a^{7}-\frac{148462322795}{772048339283}a^{6}+\frac{1261984355865}{772048339283}a^{5}-\frac{3502352669805}{772048339283}a^{4}+\frac{3415917935317}{772048339283}a^{3}-\frac{4282440835230}{772048339283}a^{2}-\frac{121097619285}{59388333791}a-\frac{74978118276}{59388333791}$, $\frac{2230458155}{772048339283}a^{9}-\frac{1297769019}{772048339283}a^{8}+\frac{53334737413}{772048339283}a^{7}-\frac{174096213}{59388333791}a^{6}+\frac{91023271310}{59388333791}a^{5}-\frac{728793262}{2047873579}a^{4}+\frac{1693378180393}{772048339283}a^{3}+\frac{2180119629544}{772048339283}a^{2}+\frac{1819769080918}{772048339283}a+\frac{50822203586}{59388333791}$, $\frac{173384857}{59388333791}a^{9}-\frac{1758168029}{772048339283}a^{8}+\frac{54524193981}{772048339283}a^{7}-\frac{7460743538}{772048339283}a^{6}+\frac{1202401920068}{772048339283}a^{5}-\frac{13343072129}{26622356527}a^{4}+\frac{2014573631582}{772048339283}a^{3}+\frac{4305819349796}{772048339283}a^{2}+\frac{2209857974276}{772048339283}a+\frac{61846303053}{59388333791}$ 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:  \( 1500.28914312 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 1500.28914312 \cdot 25}{6\cdot\sqrt{46584877058339283}}\cr\approx \mathstrut & 0.283622440523 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 25*x^8 - 10*x^7 + 552*x^6 - 313*x^5 + 1286*x^4 + 1321*x^3 + 1460*x^2 + 533*x + 169)
 
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 - x^9 + 25*x^8 - 10*x^7 + 552*x^6 - 313*x^5 + 1286*x^4 + 1321*x^3 + 1460*x^2 + 533*x + 169, 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 - x^9 + 25*x^8 - 10*x^7 + 552*x^6 - 313*x^5 + 1286*x^4 + 1321*x^3 + 1460*x^2 + 533*x + 169);
 
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 - x^9 + 25*x^8 - 10*x^7 + 552*x^6 - 313*x^5 + 1286*x^4 + 1321*x^3 + 1460*x^2 + 533*x + 169);
 
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

$C_{10}$ (as 10T1):

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 cyclic group of order 10
The 10 conjugacy class representatives for $C_{10}$
Character table for $C_{10}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.13845841.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.10.0.1}{10} }$ R ${\href{/padicField/5.10.0.1}{10} }$ ${\href{/padicField/7.5.0.1}{5} }^{2}$ ${\href{/padicField/11.2.0.1}{2} }^{5}$ ${\href{/padicField/13.1.0.1}{1} }^{10}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.10.0.1}{10} }$ ${\href{/padicField/29.2.0.1}{2} }^{5}$ ${\href{/padicField/31.5.0.1}{5} }^{2}$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.5.0.1}{5} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{5}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.10.0.1}{10} }$

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
\(3\) Copy content Toggle raw display 3.10.5.2$x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(61\) Copy content Toggle raw display 61.5.4.1$x^{5} + 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.1$x^{5} + 61$$5$$1$$4$$C_5$$[\ ]_{5}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.61.5t1.a.a$1$ $ 61 $ 5.5.13845841.1 $C_5$ (as 5T1) $0$ $1$
* 1.183.10t1.a.d$1$ $ 3 \cdot 61 $ 10.0.46584877058339283.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.61.5t1.a.b$1$ $ 61 $ 5.5.13845841.1 $C_5$ (as 5T1) $0$ $1$
* 1.183.10t1.a.a$1$ $ 3 \cdot 61 $ 10.0.46584877058339283.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.61.5t1.a.d$1$ $ 61 $ 5.5.13845841.1 $C_5$ (as 5T1) $0$ $1$
* 1.183.10t1.a.c$1$ $ 3 \cdot 61 $ 10.0.46584877058339283.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.61.5t1.a.c$1$ $ 61 $ 5.5.13845841.1 $C_5$ (as 5T1) $0$ $1$
* 1.183.10t1.a.b$1$ $ 3 \cdot 61 $ 10.0.46584877058339283.1 $C_{10}$ (as 10T1) $0$ $-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.