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Properties

Label 10.0.9630096522760791.1
Degree $10$
Signature $[0, 5]$
Discriminant $-9.630\times 10^{15}$
Root discriminant \(39.66\)
Ramified primes $3,7,11$
Class number $132$
Class group [2, 66]
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 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531)
 
gp: K = bnfinit(y^10 - y^9 + 56*y^8 - 56*y^7 + 1156*y^6 - 1156*y^5 + 10781*y^4 - 10781*y^3 + 45156*y^2 - 45156*y + 79531, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^10 - x^9 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^10 - x^9 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531)
 

\( x^{10} - x^{9} + 56 x^{8} - 56 x^{7} + 1156 x^{6} - 1156 x^{5} + 10781 x^{4} - 10781 x^{3} + \cdots + 79531 \) 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:   \(-9630096522760791\) \(\medspace = -\,3^{5}\cdot 7^{5}\cdot 11^{9}\) 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:  \(39.66\)
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:  $3^{1/2}7^{1/2}11^{9/10}\approx 39.66094555677728$
Ramified primes:   \(3\), \(7\), \(11\) 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:  \(\Q(\sqrt{-231}) \)
$\card{ \Gal(K/\Q) }$:  $10$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(231=3\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{231}(64,·)$, $\chi_{231}(1,·)$, $\chi_{231}(230,·)$, $\chi_{231}(167,·)$, $\chi_{231}(41,·)$, $\chi_{231}(83,·)$, $\chi_{231}(148,·)$, $\chi_{231}(62,·)$, $\chi_{231}(169,·)$, $\chi_{231}(190,·)$$\rbrace$
This is a CM field.
Reflex fields:  \(\Q(\sqrt{-231}) \), 10.0.9630096522760791.1$^{15}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17621}a^{6}+\frac{1287}{17621}a^{5}+\frac{30}{17621}a^{4}-\frac{3067}{17621}a^{3}+\frac{225}{17621}a^{2}+\frac{2286}{17621}a+\frac{250}{17621}$, $\frac{1}{17621}a^{7}+\frac{35}{17621}a^{5}-\frac{6435}{17621}a^{4}+\frac{350}{17621}a^{3}-\frac{5353}{17621}a^{2}+\frac{875}{17621}a-\frac{4572}{17621}$, $\frac{1}{17621}a^{8}+\frac{1383}{17621}a^{5}-\frac{700}{17621}a^{4}-\frac{3734}{17621}a^{3}-\frac{7000}{17621}a^{2}+\frac{3523}{17621}a-\frac{8750}{17621}$, $\frac{1}{17621}a^{9}-\frac{900}{17621}a^{5}+\frac{7639}{17621}a^{4}+\frac{5621}{17621}a^{3}-\frac{8095}{17621}a^{2}+\frac{1492}{17621}a+\frac{6670}{17621}$ 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

$C_{2}\times C_{66}$, which has order $132$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Relative class number: $132$

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:   $\frac{11}{17621}a^{7}+\frac{385}{17621}a^{5}-\frac{301}{17621}a^{4}+\frac{3850}{17621}a^{3}-\frac{6020}{17621}a^{2}+\frac{9625}{17621}a-\frac{15050}{17621}$, $\frac{1}{17621}a^{9}+\frac{45}{17621}a^{7}+\frac{675}{17621}a^{5}+\frac{3750}{17621}a^{3}-\frac{2286}{17621}a^{2}+\frac{5625}{17621}a-\frac{22860}{17621}$, $\frac{1}{17621}a^{9}-\frac{6}{17621}a^{8}+\frac{45}{17621}a^{7}-\frac{240}{17621}a^{6}+\frac{675}{17621}a^{5}-\frac{3000}{17621}a^{4}+\frac{4531}{17621}a^{3}-\frac{14286}{17621}a^{2}+\frac{17340}{17621}a-\frac{30360}{17621}$, $\frac{6}{17621}a^{8}+\frac{240}{17621}a^{6}+\frac{3000}{17621}a^{4}-\frac{781}{17621}a^{3}+\frac{12000}{17621}a^{2}-\frac{11715}{17621}a+\frac{7500}{17621}$ 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:  \( 26.1711060094 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 26.1711060094 \cdot 132}{2\cdot\sqrt{9630096522760791}}\cr\approx \mathstrut & 0.172365377121 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531)
 
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 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531, 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 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531);
 
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 + 56*x^8 - 56*x^7 + 1156*x^6 - 1156*x^5 + 10781*x^4 - 10781*x^3 + 45156*x^2 - 45156*x + 79531);
 
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

\(\Q(\sqrt{-231}) \), \(\Q(\zeta_{11})^+\)

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 ${\href{/padicField/2.5.0.1}{5} }^{2}$ R ${\href{/padicField/5.5.0.1}{5} }^{2}$ R R ${\href{/padicField/13.5.0.1}{5} }^{2}$ ${\href{/padicField/17.10.0.1}{10} }$ ${\href{/padicField/19.5.0.1}{5} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{5}$ ${\href{/padicField/29.5.0.1}{5} }^{2}$ ${\href{/padicField/31.10.0.1}{10} }$ ${\href{/padicField/37.5.0.1}{5} }^{2}$ ${\href{/padicField/41.10.0.1}{10} }$ ${\href{/padicField/43.2.0.1}{2} }^{5}$ ${\href{/padicField/47.5.0.1}{5} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }$ ${\href{/padicField/59.5.0.1}{5} }^{2}$

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 3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(7\) Copy content Toggle raw display 7.10.5.1$x^{10} + 2401 x^{2} - 67228$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(11\) Copy content Toggle raw display 11.10.9.1$x^{10} + 110$$10$$1$$9$$C_{10}$$[\ ]_{10}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.231.2t1.a.a$1$ $ 3 \cdot 7 \cdot 11 $ \(\Q(\sqrt{-231}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.231.10t1.b.a$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.231.10t1.b.b$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.231.10t1.b.c$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.231.10t1.b.d$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (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 *.