Properties

Label 30.10.531...753.1
Degree $30$
Signature $[10, 10]$
Discriminant $5.314\times 10^{43}$
Root discriminant $28.68$
Ramified primes $3, 7, 11$
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 - 7*x^29 + 24*x^28 - 23*x^27 - 141*x^26 + 633*x^25 - 1481*x^24 + 2556*x^23 - 3432*x^22 + 2970*x^21 + 2520*x^20 - 16434*x^19 + 36814*x^18 - 58793*x^17 + 75906*x^16 - 86484*x^15 + 87441*x^14 - 74977*x^13 + 56249*x^12 - 35409*x^11 + 19416*x^10 - 7865*x^9 + 726*x^8 + 1323*x^7 - 1451*x^6 + 413*x^5 - 168*x^4 + 17*x^3 - 15*x^2 + x + 1)
 
gp: K = bnfinit(x^30 - 7*x^29 + 24*x^28 - 23*x^27 - 141*x^26 + 633*x^25 - 1481*x^24 + 2556*x^23 - 3432*x^22 + 2970*x^21 + 2520*x^20 - 16434*x^19 + 36814*x^18 - 58793*x^17 + 75906*x^16 - 86484*x^15 + 87441*x^14 - 74977*x^13 + 56249*x^12 - 35409*x^11 + 19416*x^10 - 7865*x^9 + 726*x^8 + 1323*x^7 - 1451*x^6 + 413*x^5 - 168*x^4 + 17*x^3 - 15*x^2 + x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -15, 17, -168, 413, -1451, 1323, 726, -7865, 19416, -35409, 56249, -74977, 87441, -86484, 75906, -58793, 36814, -16434, 2520, 2970, -3432, 2556, -1481, 633, -141, -23, 24, -7, 1]);
 

\(x^{30} - 7 x^{29} + 24 x^{28} - 23 x^{27} - 141 x^{26} + 633 x^{25} - 1481 x^{24} + 2556 x^{23} - 3432 x^{22} + 2970 x^{21} + 2520 x^{20} - 16434 x^{19} + 36814 x^{18} - 58793 x^{17} + 75906 x^{16} - 86484 x^{15} + 87441 x^{14} - 74977 x^{13} + 56249 x^{12} - 35409 x^{11} + 19416 x^{10} - 7865 x^{9} + 726 x^{8} + 1323 x^{7} - 1451 x^{6} + 413 x^{5} - 168 x^{4} + 17 x^{3} - 15 x^{2} + x + 1\)  Toggle raw display

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:  \(53137573685607444656038572674510307558534753\)\(\medspace = 3^{15}\cdot 7^{10}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $28.68$
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:  $3, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{2284273730551913} a^{28} - \frac{1077711190492682}{2284273730551913} a^{27} - \frac{478250058594000}{2284273730551913} a^{26} - \frac{381912769461814}{2284273730551913} a^{25} - \frac{1075727667150217}{2284273730551913} a^{24} - \frac{31139855217457}{99316249154431} a^{23} + \frac{1052789080084286}{2284273730551913} a^{22} + \frac{1027417858945469}{2284273730551913} a^{21} - \frac{194806990134239}{2284273730551913} a^{20} - \frac{319462045477466}{2284273730551913} a^{19} - \frac{589835469961483}{2284273730551913} a^{18} - \frac{13817976646862}{99316249154431} a^{17} - \frac{346158884930264}{2284273730551913} a^{16} + \frac{1046031150596657}{2284273730551913} a^{15} - \frac{905068525821823}{2284273730551913} a^{14} - \frac{1136512461061813}{2284273730551913} a^{13} + \frac{317113488795315}{2284273730551913} a^{12} + \frac{849142985021814}{2284273730551913} a^{11} - \frac{486205425934140}{2284273730551913} a^{10} + \frac{139903097846907}{2284273730551913} a^{9} + \frac{881491547142255}{2284273730551913} a^{8} + \frac{356371297456812}{2284273730551913} a^{7} - \frac{11907816531374}{99316249154431} a^{6} + \frac{1129906232174135}{2284273730551913} a^{5} - \frac{723831734804498}{2284273730551913} a^{4} - \frac{1016607197808465}{2284273730551913} a^{3} + \frac{767116802770796}{2284273730551913} a^{2} + \frac{601755029267165}{2284273730551913} a - \frac{970762460559241}{2284273730551913}$, $\frac{1}{621936091692529210554540333929187793757950021} a^{29} - \frac{3555855602059052116379199184}{621936091692529210554540333929187793757950021} a^{28} - \frac{99843247993783861911892623269300205851945239}{621936091692529210554540333929187793757950021} a^{27} - \frac{261317288517848977577605902090769866908637343}{621936091692529210554540333929187793757950021} a^{26} + \frac{131730734950984749032190194863038778318412465}{621936091692529210554540333929187793757950021} a^{25} + \frac{181047995718165930999648315620925155356441945}{621936091692529210554540333929187793757950021} a^{24} - \frac{168643980000986818187698583982401724628388671}{621936091692529210554540333929187793757950021} a^{23} + \frac{95597114605569884268383570515901997409522292}{621936091692529210554540333929187793757950021} a^{22} + \frac{145564362251083286696491943955455962306556839}{621936091692529210554540333929187793757950021} a^{21} - \frac{60584721246564675964734739277030154318666705}{621936091692529210554540333929187793757950021} a^{20} + \frac{177990525828698058330864164470130568838601088}{621936091692529210554540333929187793757950021} a^{19} - \frac{36089622505410628708266227999177980313625562}{621936091692529210554540333929187793757950021} a^{18} - \frac{36499463799727550727650884185517961000104517}{621936091692529210554540333929187793757950021} a^{17} + \frac{92341634872123723956559953016137947022405092}{621936091692529210554540333929187793757950021} a^{16} + \frac{156908073355927188160581999239546994530963865}{621936091692529210554540333929187793757950021} a^{15} + \frac{186984814263746576073919523285256130731587769}{621936091692529210554540333929187793757950021} a^{14} + \frac{32920093210683201864754552620996904185434525}{621936091692529210554540333929187793757950021} a^{13} - \frac{14837645972907480940512088686184801700717723}{621936091692529210554540333929187793757950021} a^{12} - \frac{86887890112028826115615555713030498367812842}{621936091692529210554540333929187793757950021} a^{11} + \frac{122326489499989457419372447772165530266538427}{621936091692529210554540333929187793757950021} a^{10} + \frac{9580501869203883705413776434234138298477604}{621936091692529210554540333929187793757950021} a^{9} - \frac{132392412793637608344273081444147334023792868}{621936091692529210554540333929187793757950021} a^{8} + \frac{237345364568920067915459496889700072936285181}{621936091692529210554540333929187793757950021} a^{7} - \frac{307872510593137641038870922433677739676480093}{621936091692529210554540333929187793757950021} a^{6} + \frac{193027142264701120701644962969106971716471740}{621936091692529210554540333929187793757950021} a^{5} + \frac{99832656741118121401825312931574720977885649}{621936091692529210554540333929187793757950021} a^{4} - \frac{288212860038597664476511476857656517833021338}{621936091692529210554540333929187793757950021} a^{3} - \frac{149084831327127142049120154706484541937592904}{621936091692529210554540333929187793757950021} a^{2} + \frac{278176401371666105916278627993222076151125836}{621936091692529210554540333929187793757950021} a + \frac{133434467114462348332776503592399541066588243}{621936091692529210554540333929187793757950021}$  Toggle raw display

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 $2$)  Toggle raw display
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:  \( 26520700757.33144 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{10}\cdot(2\pi)^{10}\cdot 26520700757.33144 \cdot 1}{2\sqrt{53137573685607444656038572674510307558534753}}\approx 0.178629334289089$ (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

\(\Q(\sqrt{33}) \), 3.1.231.1, \(\Q(\zeta_{11})^+\), 6.2.1760913.1, \(\Q(\zeta_{33})^+\), 15.5.140994243189740741031.1

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

Sibling fields

Degree 30 sibling: Deg 30

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $15^{2}$ R $30$ R R $30$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ $30$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $30$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{3}$ $30$

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
3Data not computed
7Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.33.2t1.a.a$1$ $ 3 \cdot 11 $ \(\Q(\sqrt{33}) \) $C_2$ (as 2T1) $1$ $1$
1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
1.231.2t1.a.a$1$ $ 3 \cdot 7 \cdot 11 $ \(\Q(\sqrt{-231}) \) $C_2$ (as 2T1) $1$ $-1$
1.77.10t1.a.a$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
* 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) $0$ $-1$
* 1.33.10t1.b.a$1$ $ 3 \cdot 11 $ \(\Q(\zeta_{33})^+\) $C_{10}$ (as 10T1) $0$ $1$
1.231.10t1.b.b$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
1.77.10t1.a.b$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.33.10t1.b.b$1$ $ 3 \cdot 11 $ \(\Q(\zeta_{33})^+\) $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.77.10t1.a.c$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.77.10t1.a.d$1$ $ 7 \cdot 11 $ 10.0.3602729712967.1 $C_{10}$ (as 10T1) $0$ $-1$
1.231.10t1.b.c$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.33.10t1.b.c$1$ $ 3 \cdot 11 $ \(\Q(\zeta_{33})^+\) $C_{10}$ (as 10T1) $0$ $1$
1.231.10t1.b.d$1$ $ 3 \cdot 7 \cdot 11 $ 10.0.9630096522760791.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.33.10t1.b.d$1$ $ 3 \cdot 11 $ \(\Q(\zeta_{33})^+\) $C_{10}$ (as 10T1) $0$ $1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 2.231.6t3.c.a$2$ $ 3 \cdot 7 \cdot 11 $ 6.0.373527.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.231.3t2.a.a$2$ $ 3 \cdot 7 \cdot 11 $ 3.1.231.1 $S_3$ (as 3T2) $1$ $0$
* 2.2541.15t4.b.a$2$ $ 3 \cdot 7 \cdot 11^{2}$ 15.5.140994243189740741031.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.2541.15t4.b.b$2$ $ 3 \cdot 7 \cdot 11^{2}$ 15.5.140994243189740741031.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.2541.15t4.b.c$2$ $ 3 \cdot 7 \cdot 11^{2}$ 15.5.140994243189740741031.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.2541.30t12.a.a$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.10.53137573685607444656038572674510307558534753.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.2541.30t12.a.b$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.10.53137573685607444656038572674510307558534753.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.2541.30t12.a.c$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.10.53137573685607444656038572674510307558534753.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.2541.30t12.a.d$2$ $ 3 \cdot 7 \cdot 11^{2}$ 30.10.53137573685607444656038572674510307558534753.1 $C_{10}\times S_3$ (as 30T12) $0$ $0$
* 2.2541.15t4.b.d$2$ $ 3 \cdot 7 \cdot 11^{2}$ 15.5.140994243189740741031.1 $S_3 \times C_5$ (as 15T4) $0$ $0$

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 *.