Properties

Label 30.2.301...125.1
Degree $30$
Signature $[2, 14]$
Discriminant $3.013\times 10^{47}$
Root discriminant $38.25$
Ramified primes $5, 439$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1)
 
gp: K = bnfinit(x^30 - 5*x^29 + 8*x^28 + 19*x^27 - 63*x^26 - 52*x^25 + 265*x^24 - 211*x^23 - 393*x^22 + 858*x^21 - 86*x^20 - 2380*x^19 + 4186*x^18 + 8272*x^17 - 3336*x^16 - 13036*x^15 + 7176*x^14 + 23259*x^13 + 6583*x^12 - 9115*x^11 - 7328*x^10 + 1279*x^9 + 4440*x^8 + 2429*x^7 - 349*x^6 - 1189*x^5 - 88*x^4 + 271*x^3 + 100*x^2 - 13*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -13, 100, 271, -88, -1189, -349, 2429, 4440, 1279, -7328, -9115, 6583, 23259, 7176, -13036, -3336, 8272, 4186, -2380, -86, 858, -393, -211, 265, -52, -63, 19, 8, -5, 1]);
 

\( x^{30} - 5 x^{29} + 8 x^{28} + 19 x^{27} - 63 x^{26} - 52 x^{25} + 265 x^{24} - 211 x^{23} - 393 x^{22} + 858 x^{21} - 86 x^{20} - 2380 x^{19} + 4186 x^{18} + 8272 x^{17} - 3336 x^{16} - 13036 x^{15} + 7176 x^{14} + 23259 x^{13} + 6583 x^{12} - 9115 x^{11} - 7328 x^{10} + 1279 x^{9} + 4440 x^{8} + 2429 x^{7} - 349 x^{6} - 1189 x^{5} - 88 x^{4} + 271 x^{3} + 100 x^{2} - 13 x - 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(301337616246914973122131519082482771026611328125\)\(\medspace = 5^{15}\cdot 439^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $38.25$
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:  $5, 439$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{27} a^{20} + \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{4}{27} a^{12} + \frac{1}{27} a^{11} - \frac{4}{27} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} + \frac{5}{27} a^{7} - \frac{4}{27} a^{6} + \frac{5}{27} a^{5} + \frac{1}{3} a^{4} - \frac{13}{27} a^{3} - \frac{8}{27} a - \frac{8}{27}$, $\frac{1}{27} a^{21} + \frac{1}{27} a^{19} - \frac{1}{27} a^{18} - \frac{1}{27} a^{17} + \frac{1}{27} a^{16} + \frac{1}{27} a^{15} + \frac{1}{27} a^{14} - \frac{4}{27} a^{13} + \frac{1}{27} a^{12} - \frac{4}{27} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{27} a^{8} - \frac{4}{27} a^{7} + \frac{5}{27} a^{6} + \frac{1}{3} a^{5} - \frac{13}{27} a^{4} - \frac{8}{27} a^{2} - \frac{8}{27} a + \frac{1}{3}$, $\frac{1}{27} a^{22} - \frac{1}{27} a^{19} + \frac{1}{27} a^{18} - \frac{1}{27} a^{17} - \frac{1}{27} a^{16} + \frac{4}{27} a^{14} - \frac{4}{27} a^{11} + \frac{4}{27} a^{10} - \frac{4}{27} a^{9} - \frac{4}{27} a^{8} + \frac{4}{27} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{5}{27} a^{3} + \frac{13}{27} a^{2} - \frac{4}{27} a - \frac{13}{27}$, $\frac{1}{27} a^{23} + \frac{1}{27} a^{19} + \frac{1}{27} a^{17} - \frac{1}{27} a^{16} - \frac{1}{27} a^{15} + \frac{4}{27} a^{14} + \frac{4}{27} a^{13} + \frac{4}{27} a^{12} - \frac{1}{27} a^{11} + \frac{4}{27} a^{10} - \frac{1}{27} a^{9} + \frac{2}{9} a^{7} - \frac{7}{27} a^{6} + \frac{11}{27} a^{5} - \frac{7}{27} a^{4} - \frac{1}{9} a^{3} + \frac{2}{27} a^{2} - \frac{4}{9} a + \frac{7}{27}$, $\frac{1}{27} a^{24} - \frac{1}{9} a^{8} + \frac{2}{27}$, $\frac{1}{27} a^{25} - \frac{1}{9} a^{9} + \frac{2}{27} a$, $\frac{1}{81} a^{26} + \frac{1}{81} a^{25} - \frac{1}{81} a^{22} - \frac{2}{81} a^{19} - \frac{1}{81} a^{18} + \frac{4}{81} a^{17} - \frac{2}{81} a^{16} - \frac{1}{27} a^{15} - \frac{7}{81} a^{14} + \frac{2}{27} a^{13} + \frac{2}{27} a^{12} - \frac{11}{81} a^{11} - \frac{10}{81} a^{10} + \frac{1}{81} a^{9} - \frac{11}{81} a^{8} + \frac{13}{27} a^{7} + \frac{8}{81} a^{6} - \frac{11}{27} a^{5} - \frac{5}{27} a^{4} - \frac{32}{81} a^{3} + \frac{1}{81} a^{2} + \frac{4}{27} a + \frac{13}{81}$, $\frac{1}{729} a^{27} + \frac{2}{729} a^{26} + \frac{10}{729} a^{25} - \frac{4}{243} a^{24} + \frac{11}{729} a^{23} - \frac{7}{729} a^{22} + \frac{1}{81} a^{21} + \frac{7}{729} a^{20} + \frac{11}{243} a^{19} - \frac{4}{243} a^{18} + \frac{29}{729} a^{17} - \frac{38}{729} a^{16} - \frac{13}{729} a^{15} + \frac{86}{729} a^{14} - \frac{10}{243} a^{13} - \frac{101}{729} a^{12} + \frac{2}{81} a^{11} - \frac{40}{243} a^{10} - \frac{115}{729} a^{9} + \frac{43}{729} a^{8} + \frac{326}{729} a^{7} + \frac{182}{729} a^{6} - \frac{80}{243} a^{5} - \frac{131}{729} a^{4} - \frac{88}{729} a^{3} - \frac{167}{729} a^{2} + \frac{220}{729} a - \frac{47}{729}$, $\frac{1}{384183} a^{28} - \frac{260}{384183} a^{27} - \frac{847}{384183} a^{26} - \frac{2965}{384183} a^{25} - \frac{112}{384183} a^{24} + \frac{1}{279} a^{23} - \frac{4088}{384183} a^{22} + \frac{1132}{384183} a^{21} + \frac{5597}{384183} a^{20} + \frac{448}{14229} a^{19} + \frac{18815}{384183} a^{18} - \frac{14665}{384183} a^{17} - \frac{20954}{384183} a^{16} + \frac{334}{42687} a^{15} + \frac{53290}{384183} a^{14} + \frac{60436}{384183} a^{13} - \frac{29842}{384183} a^{12} + \frac{18671}{128061} a^{11} + \frac{16835}{384183} a^{10} + \frac{58487}{384183} a^{9} - \frac{35114}{384183} a^{8} + \frac{12025}{42687} a^{7} + \frac{85303}{384183} a^{6} + \frac{11051}{22599} a^{5} - \frac{166322}{384183} a^{4} + \frac{7787}{384183} a^{3} + \frac{10474}{42687} a^{2} - \frac{52430}{128061} a - \frac{56923}{384183}$, $\frac{1}{6932787919812818813867113376921047991049} a^{29} - \frac{2153464308234134186111767009577}{2760966913505702434833577609287553959} a^{28} + \frac{970866685316710843053001252395220706}{6932787919812818813867113376921047991049} a^{27} - \frac{33093227870933222437200936937984443946}{6932787919812818813867113376921047991049} a^{26} - \frac{63260015356179361793332499850241201937}{6932787919812818813867113376921047991049} a^{25} + \frac{112786505585563284222044559681333556342}{6932787919812818813867113376921047991049} a^{24} - \frac{40306421851876579696840883628992433827}{2310929306604272937955704458973682663683} a^{23} + \frac{117398551423359694093899420270077188538}{6932787919812818813867113376921047991049} a^{22} - \frac{50740576993151739325591307068645201112}{6932787919812818813867113376921047991049} a^{21} - \frac{7101360271869078289308228435849127367}{407811054106636400815712551583591058297} a^{20} + \frac{298686993682945136523790333949380241543}{6932787919812818813867113376921047991049} a^{19} + \frac{41670478074274955750636118241830360332}{2310929306604272937955704458973682663683} a^{18} + \frac{160629448965266180846727285419116589542}{6932787919812818813867113376921047991049} a^{17} - \frac{128139266846633417657081684006436781891}{2310929306604272937955704458973682663683} a^{16} + \frac{13204749581417629795335256186484005111}{770309768868090979318568152991227554561} a^{15} - \frac{292584915461408707535096495633993477974}{6932787919812818813867113376921047991049} a^{14} - \frac{1126925107580585852878934794282464997514}{6932787919812818813867113376921047991049} a^{13} + \frac{59366859038605883876112624759289008911}{6932787919812818813867113376921047991049} a^{12} + \frac{195349447251961042896658894096099465856}{6932787919812818813867113376921047991049} a^{11} + \frac{14575708550595851788280901405743425432}{135937018035545466938570850527863686099} a^{10} + \frac{198696386812724344793597817304947872731}{6932787919812818813867113376921047991049} a^{9} + \frac{31383265940718125495287212902493520715}{770309768868090979318568152991227554561} a^{8} + \frac{322087799195464009567506905642194300139}{2310929306604272937955704458973682663683} a^{7} - \frac{1860244480352380588947016078422033983488}{6932787919812818813867113376921047991049} a^{6} + \frac{304042728037087395132654496874621507314}{770309768868090979318568152991227554561} a^{5} + \frac{1547545954021488284910428209144879051637}{6932787919812818813867113376921047991049} a^{4} - \frac{425515906131768765781378707840133084640}{2310929306604272937955704458973682663683} a^{3} + \frac{1778951606093810719888709702795166272038}{6932787919812818813867113376921047991049} a^{2} - \frac{96591288373017859824994113018363798506}{256769922956030326439522717663742518187} a + \frac{1624677260981336965511385422630217276962}{6932787919812818813867113376921047991049}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $15$
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:  \( 748323688575.9082 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 748323688575.9082 \cdot 2}{2\sqrt{301337616246914973122131519082482771026611328125}}\approx 0.814968771317785$ (assuming GRH)

Galois group

$D_{30}$ (as 30T14):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.439.1, 5.1.192721.1, 6.2.24090125.1, 10.2.116066824503125.1, 15.1.3142328914862177479.1

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 $30$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{15}$ R $30$ $15^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{15}$ $30$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
439Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2195.2t1.a.a$1$ $ 5 \cdot 439 $ \(\Q(\sqrt{-2195}) \) $C_2$ (as 2T1) $1$ $-1$
1.439.2t1.a.a$1$ $ 439 $ \(\Q(\sqrt{-439}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
* 2.10975.6t3.a.a$2$ $ 5^{2} \cdot 439 $ 6.0.10575564875.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.439.3t2.a.a$2$ $ 439 $ 3.1.439.1 $S_3$ (as 3T2) $1$ $0$
* 2.439.5t2.a.a$2$ $ 439 $ 5.1.192721.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.439.5t2.a.b$2$ $ 439 $ 5.1.192721.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.10975.10t3.a.b$2$ $ 5^{2} \cdot 439 $ 10.0.50953335956871875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.10975.10t3.a.a$2$ $ 5^{2} \cdot 439 $ 10.0.50953335956871875.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.439.15t2.a.a$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.10975.30t14.a.c$2$ $ 5^{2} \cdot 439 $ 30.2.301337616246914973122131519082482771026611328125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.439.15t2.a.c$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.10975.30t14.a.a$2$ $ 5^{2} \cdot 439 $ 30.2.301337616246914973122131519082482771026611328125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.439.15t2.a.b$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.10975.30t14.a.b$2$ $ 5^{2} \cdot 439 $ 30.2.301337616246914973122131519082482771026611328125.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.439.15t2.a.d$2$ $ 439 $ 15.1.3142328914862177479.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.10975.30t14.a.d$2$ $ 5^{2} \cdot 439 $ 30.2.301337616246914973122131519082482771026611328125.1 $D_{30}$ (as 30T14) $1$ $0$

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 *.