Properties

Label 30.0.457...771.1
Degree $30$
Signature $[0, 15]$
Discriminant $-4.571\times 10^{37}$
Root discriminant $18.00$
Ramified primes $3, 11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{15}\times S_3$ (as 30T15)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^30 + 6*x^28 - x^27 + 14*x^26 - x^25 + 8*x^24 + 13*x^23 - 39*x^22 + 37*x^21 - 93*x^20 + 30*x^19 + 10*x^18 - 13*x^17 + 349*x^16 + 77*x^15 + 721*x^14 + 575*x^13 + 1026*x^12 + 1233*x^11 + 1137*x^10 + 993*x^9 + 501*x^8 - 41*x^7 - 253*x^6 - 186*x^5 - 58*x^4 + 6*x^3 + 14*x^2 + 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 14, 6, -58, -186, -253, -41, 501, 993, 1137, 1233, 1026, 575, 721, 77, 349, -13, 10, 30, -93, 37, -39, 13, 8, -1, 14, -1, 6, 0, 1]);
 

\( x^{30} + 6 x^{28} - x^{27} + 14 x^{26} - x^{25} + 8 x^{24} + 13 x^{23} - 39 x^{22} + 37 x^{21} - 93 x^{20} + 30 x^{19} + 10 x^{18} - 13 x^{17} + 349 x^{16} + 77 x^{15} + 721 x^{14} + 575 x^{13} + 1026 x^{12} + 1233 x^{11} + 1137 x^{10} + 993 x^{9} + 501 x^{8} - 41 x^{7} - 253 x^{6} - 186 x^{5} - 58 x^{4} + 6 x^{3} + 14 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-45711723244122563996456104229882472771\)\(\medspace = -\,3^{20}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $18.00$
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:  $3, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $15$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{445} a^{25} + \frac{11}{445} a^{24} - \frac{219}{445} a^{23} + \frac{86}{445} a^{22} + \frac{39}{89} a^{21} + \frac{186}{445} a^{20} + \frac{82}{445} a^{19} - \frac{26}{89} a^{18} + \frac{203}{445} a^{17} + \frac{153}{445} a^{16} + \frac{167}{445} a^{15} + \frac{169}{445} a^{14} + \frac{23}{89} a^{13} - \frac{34}{89} a^{12} + \frac{51}{445} a^{11} - \frac{167}{445} a^{10} - \frac{164}{445} a^{9} - \frac{25}{89} a^{8} - \frac{10}{89} a^{7} + \frac{84}{445} a^{6} - \frac{4}{445} a^{5} + \frac{212}{445} a^{4} + \frac{18}{89} a^{3} - \frac{32}{89} a^{2} + \frac{18}{445} a + \frac{152}{445}$, $\frac{1}{445} a^{26} + \frac{21}{89} a^{24} - \frac{35}{89} a^{23} + \frac{139}{445} a^{22} - \frac{179}{445} a^{21} - \frac{184}{445} a^{20} - \frac{142}{445} a^{19} - \frac{147}{445} a^{18} + \frac{29}{89} a^{17} - \frac{181}{445} a^{16} + \frac{112}{445} a^{15} + \frac{36}{445} a^{14} - \frac{20}{89} a^{13} + \frac{141}{445} a^{12} + \frac{162}{445} a^{11} - \frac{107}{445} a^{10} - \frac{101}{445} a^{9} - \frac{2}{89} a^{8} + \frac{189}{445} a^{7} - \frac{38}{445} a^{6} - \frac{189}{445} a^{5} - \frac{17}{445} a^{4} + \frac{37}{89} a^{3} - \frac{2}{445} a^{2} - \frac{46}{445} a + \frac{108}{445}$, $\frac{1}{445} a^{27} + \frac{1}{89} a^{24} - \frac{6}{445} a^{23} + \frac{136}{445} a^{22} - \frac{189}{445} a^{21} - \frac{92}{445} a^{20} + \frac{143}{445} a^{19} - \frac{136}{445} a^{17} + \frac{67}{445} a^{16} - \frac{144}{445} a^{15} - \frac{9}{89} a^{14} + \frac{81}{445} a^{13} + \frac{212}{445} a^{12} - \frac{122}{445} a^{11} + \frac{79}{445} a^{10} - \frac{29}{89} a^{9} - \frac{36}{445} a^{8} - \frac{128}{445} a^{7} - \frac{109}{445} a^{6} - \frac{42}{445} a^{5} + \frac{35}{89} a^{4} - \frac{107}{445} a^{3} - \frac{156}{445} a^{2} - \frac{2}{445} a + \frac{12}{89}$, $\frac{1}{13387012429555} a^{28} + \frac{11292184506}{13387012429555} a^{27} + \frac{239979340}{2677402485911} a^{26} - \frac{11621511658}{13387012429555} a^{25} - \frac{4902792104044}{13387012429555} a^{24} - \frac{6166493836393}{13387012429555} a^{23} - \frac{5758271901301}{13387012429555} a^{22} - \frac{1341906105356}{13387012429555} a^{21} - \frac{5466840494667}{13387012429555} a^{20} + \frac{6444606444167}{13387012429555} a^{19} + \frac{6175612326114}{13387012429555} a^{18} - \frac{1231175297543}{13387012429555} a^{17} + \frac{1579876048774}{13387012429555} a^{16} - \frac{896338943675}{2677402485911} a^{15} + \frac{4836948846369}{13387012429555} a^{14} + \frac{4601689343788}{13387012429555} a^{13} - \frac{511854111159}{2677402485911} a^{12} - \frac{5347110096196}{13387012429555} a^{11} - \frac{991991156498}{2677402485911} a^{10} - \frac{2241961602974}{13387012429555} a^{9} + \frac{2053307832501}{13387012429555} a^{8} - \frac{759610988112}{13387012429555} a^{7} + \frac{4602001636602}{13387012429555} a^{6} - \frac{848191643384}{2677402485911} a^{5} - \frac{606966174518}{13387012429555} a^{4} - \frac{1820764963473}{13387012429555} a^{3} + \frac{589929716967}{13387012429555} a^{2} - \frac{3254376689291}{13387012429555} a - \frac{2112461484861}{13387012429555}$, $\frac{1}{13387012429555} a^{29} + \frac{1002026522}{2677402485911} a^{27} + \frac{14689278362}{13387012429555} a^{26} + \frac{13605534833}{13387012429555} a^{25} + \frac{324983930549}{2677402485911} a^{24} + \frac{682285836191}{2677402485911} a^{23} - \frac{1164366560091}{2677402485911} a^{22} - \frac{1045981908}{2677402485911} a^{21} + \frac{4249606413601}{13387012429555} a^{20} + \frac{3656525880493}{13387012429555} a^{19} + \frac{1039894128888}{13387012429555} a^{18} - \frac{5903150631877}{13387012429555} a^{17} - \frac{326707186607}{2677402485911} a^{16} - \frac{3339284003187}{13387012429555} a^{15} + \frac{1170255526547}{2677402485911} a^{14} - \frac{6669905464812}{13387012429555} a^{13} + \frac{6663867702186}{13387012429555} a^{12} + \frac{3666619169923}{13387012429555} a^{11} - \frac{2379384253328}{13387012429555} a^{10} - \frac{4240524289601}{13387012429555} a^{9} - \frac{1682088816724}{13387012429555} a^{8} + \frac{293229730896}{13387012429555} a^{7} + \frac{451143399178}{2677402485911} a^{6} + \frac{2876129479224}{13387012429555} a^{5} + \frac{1849453151018}{13387012429555} a^{4} + \frac{604978445978}{13387012429555} a^{3} - \frac{2196819749374}{13387012429555} a^{2} + \frac{883452233921}{2677402485911} a - \frac{2116054177081}{13387012429555}$

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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{12653957152607}{2677402485911} a^{29} - \frac{6159480532392}{2677402485911} a^{28} + \frac{78477429740129}{2677402485911} a^{27} - \frac{50349640529454}{2677402485911} a^{26} + \frac{198772937530862}{2677402485911} a^{25} - \frac{105849092098559}{2677402485911} a^{24} + \frac{144663718380552}{2677402485911} a^{23} + \frac{102353127789123}{2677402485911} a^{22} - \frac{550735971791860}{2677402485911} a^{21} + \frac{736058773339872}{2677402485911} a^{20} - \frac{1513402505071755}{2677402485911} a^{19} + \frac{1078104915700476}{2677402485911} a^{18} - \frac{328577462792968}{2677402485911} a^{17} - \frac{76704618826301}{2677402485911} a^{16} + \frac{4489202749714251}{2677402485911} a^{15} - \frac{1215995815292423}{2677402485911} a^{14} + \frac{9556496914311311}{2677402485911} a^{13} + \frac{2768316816085662}{2677402485911} a^{12} + \frac{11274958798156982}{2677402485911} a^{11} + \frac{10235722776091104}{2677402485911} a^{10} + \frac{9077290907100770}{2677402485911} a^{9} + \frac{8052604766471538}{2677402485911} a^{8} + \frac{2342098007255521}{2677402485911} a^{7} - \frac{1717269073290881}{2677402485911} a^{6} - \frac{2251867087989700}{2677402485911} a^{5} - \frac{1126642969502881}{2677402485911} a^{4} - \frac{133995717584920}{2677402485911} a^{3} + \frac{136551788434983}{2677402485911} a^{2} + \frac{93114123118404}{2677402485911} a + \frac{22939094334598}{2677402485911} \) (order $22$)
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:  \( 27376181.549941503 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 27376181.549941503 \cdot 1}{22\sqrt{45711723244122563996456104229882472771}}\approx 0.172836084881715$ (assuming GRH)

Galois group

$C_{15}\times S_3$ (as 30T15):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 90
The 45 conjugacy class representatives for $C_{15}\times S_3$
Character table for $C_{15}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), 6.0.107811.1, \(\Q(\zeta_{11})\)

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

Sibling fields

Degree 45 sibling: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $30$ R $15{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{3}$ $30$ R $30$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ $30$ $15{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{3}$ $15^{2}$ $30$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ $15{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{3}$ $15^{2}$ $15{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{3}$

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
$3$3.15.20.99$x^{15} + 42 x^{14} + 27 x^{13} + 31 x^{12} + 45 x^{11} + 45 x^{10} + 41 x^{9} + 30 x^{8} + 42 x^{7} + 8 x^{6} + 75 x^{5} + 18 x^{4} + 42 x^{3} + 24 x^{2} + 51 x + 19$$3$$5$$20$$C_{15}$$[2]^{5}$
3.15.0.1$x^{15} + x^{2} - x + 1$$1$$15$$0$$C_{15}$$[\ ]^{15}$
11Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.11.2t1.a.a$1$ $ 11 $ \(\Q(\sqrt{-11}) \) $C_2$ (as 2T1) $1$ $-1$
1.99.6t1.a.a$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
1.99.6t1.a.b$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.11.10t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.10t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
1.99.30t1.a.a$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.30t1.a.b$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.15t1.a.a$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.c$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.30t1.a.d$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.15t1.a.b$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.e$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.15t1.a.c$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.f$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.15t1.a.d$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.g$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.15t1.a.e$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.30t1.a.h$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
1.99.15t1.a.f$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.15t1.a.g$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
1.99.15t1.a.h$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
2.891.3t2.b.a$2$ $ 3^{4} \cdot 11 $ 3.1.891.1 $S_3$ (as 3T2) $1$ $0$
* 2.99.6t5.a.a$2$ $ 3^{2} \cdot 11 $ 6.0.107811.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.99.6t5.a.b$2$ $ 3^{2} \cdot 11 $ 6.0.107811.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.9801.15t4.a.a$2$ $ 3^{4} \cdot 11^{2}$ 15.5.120373254183671877620331.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
2.9801.15t4.a.b$2$ $ 3^{4} \cdot 11^{2}$ 15.5.120373254183671877620331.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
2.9801.15t4.a.c$2$ $ 3^{4} \cdot 11^{2}$ 15.5.120373254183671877620331.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
2.9801.15t4.a.d$2$ $ 3^{4} \cdot 11^{2}$ 15.5.120373254183671877620331.1 $S_3 \times C_5$ (as 15T4) $0$ $0$
* 2.1089.30t15.a.a$2$ $ 3^{2} \cdot 11^{2}$ 30.0.45711723244122563996456104229882472771.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.1089.30t15.a.b$2$ $ 3^{2} \cdot 11^{2}$ 30.0.45711723244122563996456104229882472771.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.1089.30t15.a.c$2$ $ 3^{2} \cdot 11^{2}$ 30.0.45711723244122563996456104229882472771.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.1089.30t15.a.d$2$ $ 3^{2} \cdot 11^{2}$ 30.0.45711723244122563996456104229882472771.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.1089.30t15.a.e$2$ $ 3^{2} \cdot 11^{2}$ 30.0.45711723244122563996456104229882472771.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.1089.30t15.a.f$2$ $ 3^{2} \cdot 11^{2}$ 30.0.45711723244122563996456104229882472771.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.1089.30t15.a.g$2$ $ 3^{2} \cdot 11^{2}$ 30.0.45711723244122563996456104229882472771.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$
* 2.1089.30t15.a.h$2$ $ 3^{2} \cdot 11^{2}$ 30.0.45711723244122563996456104229882472771.1 $C_{15}\times S_3$ (as 30T15) $0$ $0$

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 *.