Properties

Label 30.10.144...896.1
Degree $30$
Signature $[10, 10]$
Discriminant $1.441\times 10^{40}$
Root discriminant $21.81$
Ramified primes $2, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{10}\times S_3$ (as 30T12)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11)
 
gp: K = bnfinit(x^30 - 15*x^28 + 104*x^26 - 482*x^24 + 1708*x^22 - 4753*x^20 + 10608*x^18 - 18925*x^16 + 26563*x^14 - 28460*x^12 + 22463*x^10 - 12716*x^8 + 5049*x^6 - 1331*x^4 + 198*x^2 - 11, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 198, 0, -1331, 0, 5049, 0, -12716, 0, 22463, 0, -28460, 0, 26563, 0, -18925, 0, 10608, 0, -4753, 0, 1708, 0, -482, 0, 104, 0, -15, 0, 1]);
 

\( x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[10, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(14414591053629993500878399466353926864896\)\(\medspace = 2^{40}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.81$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $10$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{26} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{27} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{587409483542016518} a^{28} - \frac{103020836976652849}{587409483542016518} a^{26} - \frac{6816620393891934}{293704741771008259} a^{24} + \frac{149570912943989}{5389077830660702} a^{22} - \frac{47659317464097223}{293704741771008259} a^{20} + \frac{8081375902375867}{587409483542016518} a^{18} + \frac{123471128679091941}{587409483542016518} a^{16} - \frac{16960726100570672}{293704741771008259} a^{14} + \frac{126667923291359885}{587409483542016518} a^{12} - \frac{1}{2} a^{11} - \frac{76716716275769853}{587409483542016518} a^{10} - \frac{1}{2} a^{9} - \frac{237814649392225551}{587409483542016518} a^{8} - \frac{1}{2} a^{7} - \frac{127991483737504008}{293704741771008259} a^{6} - \frac{1}{2} a^{5} - \frac{143232585408898121}{293704741771008259} a^{4} - \frac{55036218279611367}{587409483542016518} a^{2} - \frac{1}{2} a + \frac{208974501016961141}{587409483542016518}$, $\frac{1}{587409483542016518} a^{29} - \frac{103020836976652849}{587409483542016518} a^{27} - \frac{6816620393891934}{293704741771008259} a^{25} + \frac{149570912943989}{5389077830660702} a^{23} - \frac{47659317464097223}{293704741771008259} a^{21} + \frac{8081375902375867}{587409483542016518} a^{19} + \frac{123471128679091941}{587409483542016518} a^{17} - \frac{16960726100570672}{293704741771008259} a^{15} + \frac{126667923291359885}{587409483542016518} a^{13} - \frac{1}{2} a^{12} - \frac{76716716275769853}{587409483542016518} a^{11} - \frac{1}{2} a^{10} - \frac{237814649392225551}{587409483542016518} a^{9} - \frac{1}{2} a^{8} - \frac{127991483737504008}{293704741771008259} a^{7} - \frac{1}{2} a^{6} - \frac{143232585408898121}{293704741771008259} a^{5} - \frac{55036218279611367}{587409483542016518} a^{3} - \frac{1}{2} a^{2} + \frac{208974501016961141}{587409483542016518} a$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $19$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 397294072.2356694 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{10}\cdot(2\pi)^{10}\cdot 397294072.2356694 \cdot 1}{2\sqrt{14414591053629993500878399466353926864896}}\approx 0.162472389195522$ (assuming GRH)

Galois group

$C_{10}\times S_3$ (as 30T12):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 30 conjugacy class representatives for $C_{10}\times S_3$
Character table for $C_{10}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), 3.1.44.1, \(\Q(\zeta_{11})^+\), 6.2.340736.1, \(\Q(\zeta_{44})^+\), 15.5.35351257235385344.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 sibling: 30.0.1310417368511817590988945406032175169536.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $30$ $15^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ $30$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ $30$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
11Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.4.2t1.a.a$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.44.2t1.a.a$1$ $ 2^{2} \cdot 11 $ $x^{2} - 11$ $C_2$ (as 2T1) $1$ $1$
1.11.2t1.a.a$1$ $ 11 $ $x^{2} - x + 3$ $C_2$ (as 2T1) $1$ $-1$
* 1.44.10t1.b.a$1$ $ 2^{2} \cdot 11 $ $x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11$ $C_{10}$ (as 10T1) $0$ $1$
1.44.10t1.a.a$1$ $ 2^{2} \cdot 11 $ $x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$ $C_{10}$ (as 10T1) $0$ $-1$
1.11.10t1.a.a$1$ $ 11 $ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.a$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.11.10t1.a.b$1$ $ 11 $ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.44.10t1.b.b$1$ $ 2^{2} \cdot 11 $ $x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11$ $C_{10}$ (as 10T1) $0$ $1$
1.11.10t1.a.c$1$ $ 11 $ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{10}$ (as 10T1) $0$ $-1$
1.44.10t1.a.b$1$ $ 2^{2} \cdot 11 $ $x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$ $C_{10}$ (as 10T1) $0$ $-1$
1.11.10t1.a.d$1$ $ 11 $ $x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.b$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.c$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
* 1.44.10t1.b.c$1$ $ 2^{2} \cdot 11 $ $x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11$ $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.d$1$ $ 11 $ $x^{5} - x^{4} - 4 x^{3} + 3 x^{2} + 3 x - 1$ $C_5$ (as 5T1) $0$ $1$
1.44.10t1.a.c$1$ $ 2^{2} \cdot 11 $ $x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$ $C_{10}$ (as 10T1) $0$ $-1$
1.44.10t1.a.d$1$ $ 2^{2} \cdot 11 $ $x^{10} + 9 x^{8} + 28 x^{6} + 35 x^{4} + 15 x^{2} + 1$ $C_{10}$ (as 10T1) $0$ $-1$
* 1.44.10t1.b.d$1$ $ 2^{2} \cdot 11 $ $x^{10} - 11 x^{8} + 44 x^{6} - 77 x^{4} + 55 x^{2} - 11$ $C_{10}$ (as 10T1) $0$ $1$
* 2.176.6t3.b.a$2$ $ 2^{4} \cdot 11 $ $x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 4 x^{2} - 4 x + 2$ $D_{6}$ (as 6T3) $1$ $0$
* 2.44.3t2.b.a$2$ $ 2^{2} \cdot 11 $ $x^{3} - x^{2} + x + 1$ $S_3$ (as 3T2) $1$ $0$
* 2.484.15t4.a.a$2$ $ 2^{2} \cdot 11^{2}$ $x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1$ $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.484.15t4.a.b$2$ $ 2^{2} \cdot 11^{2}$ $x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1$ $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.484.15t4.a.c$2$ $ 2^{2} \cdot 11^{2}$ $x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1$ $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.1936.30t12.b.a$2$ $ 2^{4} \cdot 11^{2}$ $x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11$ $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.1936.30t12.b.b$2$ $ 2^{4} \cdot 11^{2}$ $x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11$ $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.1936.30t12.b.c$2$ $ 2^{4} \cdot 11^{2}$ $x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11$ $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.484.15t4.a.d$2$ $ 2^{2} \cdot 11^{2}$ $x^{15} - 3 x^{13} - 5 x^{12} + 8 x^{11} + x^{10} + 14 x^{9} - 13 x^{8} - 17 x^{7} + 4 x^{6} + x^{5} + 22 x^{4} - 2 x^{3} - 8 x^{2} - 3 x - 1$ $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.1936.30t12.b.d$2$ $ 2^{4} \cdot 11^{2}$ $x^{30} - 15 x^{28} + 104 x^{26} - 482 x^{24} + 1708 x^{22} - 4753 x^{20} + 10608 x^{18} - 18925 x^{16} + 26563 x^{14} - 28460 x^{12} + 22463 x^{10} - 12716 x^{8} + 5049 x^{6} - 1331 x^{4} + 198 x^{2} - 11$ $C_{10}\times S_3$ (as 30T12) $0$ $0$

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 *.