Properties

Label 30.0.159...171.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.594\times 10^{47}$
Root discriminant $37.45$
Ramified primes $3, 11$
Class number $93$ (GRH)
Class group $[93]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^30 + 3*x^28 - x^27 + 9*x^26 - 6*x^25 + 28*x^24 - 27*x^23 + 90*x^22 - 109*x^21 + 297*x^20 + 507*x^19 + 1000*x^18 + 1224*x^17 + 2493*x^16 + 2672*x^15 + 6255*x^14 + 5523*x^13 + 16093*x^12 + 10314*x^11 + 42756*x^10 + 14849*x^9 + 5157*x^8 + 1791*x^7 + 622*x^6 + 216*x^5 + 75*x^4 + 26*x^3 + 9*x^2 + 3*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 9, 26, 75, 216, 622, 1791, 5157, 14849, 42756, 10314, 16093, 5523, 6255, 2672, 2493, 1224, 1000, 507, 297, -109, 90, -27, 28, -6, 9, -1, 3, 0, 1]);
 

\( x^{30} + 3 x^{28} - x^{27} + 9 x^{26} - 6 x^{25} + 28 x^{24} - 27 x^{23} + 90 x^{22} - 109 x^{21} + 297 x^{20} + 507 x^{19} + 1000 x^{18} + 1224 x^{17} + 2493 x^{16} + 2672 x^{15} + 6255 x^{14} + 5523 x^{13} + 16093 x^{12} + 10314 x^{11} + 42756 x^{10} + 14849 x^{9} + 5157 x^{8} + 1791 x^{7} + 622 x^{6} + 216 x^{5} + 75 x^{4} + 26 x^{3} + 9 x^{2} + 3 x + 1 \)

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:  \(-159386923550435671074967363509984324121230045171\)\(\medspace = -\,3^{40}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $37.45$
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)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois and abelian over $\Q$.
Conductor:  \(99=3^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{99}(64,·)$, $\chi_{99}(1,·)$, $\chi_{99}(67,·)$, $\chi_{99}(4,·)$, $\chi_{99}(70,·)$, $\chi_{99}(7,·)$, $\chi_{99}(73,·)$, $\chi_{99}(10,·)$, $\chi_{99}(76,·)$, $\chi_{99}(13,·)$, $\chi_{99}(79,·)$, $\chi_{99}(16,·)$, $\chi_{99}(82,·)$, $\chi_{99}(19,·)$, $\chi_{99}(85,·)$, $\chi_{99}(25,·)$, $\chi_{99}(91,·)$, $\chi_{99}(28,·)$, $\chi_{99}(94,·)$, $\chi_{99}(31,·)$, $\chi_{99}(97,·)$, $\chi_{99}(34,·)$, $\chi_{99}(37,·)$, $\chi_{99}(40,·)$, $\chi_{99}(43,·)$, $\chi_{99}(46,·)$, $\chi_{99}(49,·)$, $\chi_{99}(52,·)$, $\chi_{99}(58,·)$, $\chi_{99}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{14317073} a^{21} - \frac{4597228}{14317073} a^{20} + \frac{48209}{14317073} a^{19} + \frac{525388}{14317073} a^{18} + \frac{4741855}{14317073} a^{17} + \frac{1527955}{14317073} a^{16} - \frac{616896}{14317073} a^{15} - \frac{157990}{14317073} a^{14} - \frac{3378643}{14317073} a^{13} + \frac{142926}{14317073} a^{12} + \frac{4339134}{14317073} a^{11} - \frac{5254364}{14317073} a^{10} + \frac{1444706}{14317073} a^{9} + \frac{4338513}{14317073} a^{8} - \frac{4728591}{14317073} a^{7} - \frac{2746240}{14317073} a^{6} - \frac{4207213}{14317073} a^{5} - \frac{3510129}{14317073} a^{4} + \frac{4441674}{14317073} a^{3} - \frac{6323174}{14317073} a^{2} + \frac{2518078}{14317073} a + \frac{5222950}{14317073}$, $\frac{1}{14317073} a^{22} + \frac{5255288}{14317073} a^{11} + \frac{439665}{14317073}$, $\frac{1}{14317073} a^{23} + \frac{5255288}{14317073} a^{12} + \frac{439665}{14317073} a$, $\frac{1}{14317073} a^{24} + \frac{5255288}{14317073} a^{13} + \frac{439665}{14317073} a^{2}$, $\frac{1}{14317073} a^{25} + \frac{5255288}{14317073} a^{14} + \frac{439665}{14317073} a^{3}$, $\frac{1}{14317073} a^{26} + \frac{5255288}{14317073} a^{15} + \frac{439665}{14317073} a^{4}$, $\frac{1}{14317073} a^{27} + \frac{5255288}{14317073} a^{16} + \frac{439665}{14317073} a^{5}$, $\frac{1}{14317073} a^{28} + \frac{5255288}{14317073} a^{17} + \frac{439665}{14317073} a^{6}$, $\frac{1}{14317073} a^{29} + \frac{5255288}{14317073} a^{18} + \frac{439665}{14317073} a^{7}$

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

Class group and class number

$C_{93}$, which has order $93$ (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:  \( -\frac{1799175}{14317073} a^{28} + \frac{599725}{14317073} a^{27} - \frac{5397525}{14317073} a^{26} + \frac{3598350}{14317073} a^{25} - \frac{16792300}{14317073} a^{24} + \frac{16192575}{14317073} a^{23} - \frac{53975250}{14317073} a^{22} + \frac{65370025}{14317073} a^{21} - \frac{178118325}{14317073} a^{20} + \frac{250084998}{14317073} a^{19} - \frac{599725000}{14317073} a^{18} - \frac{734063400}{14317073} a^{17} - \frac{1495114425}{14317073} a^{16} - \frac{1602465200}{14317073} a^{15} - \frac{3751279875}{14317073} a^{14} - \frac{3312281175}{14317073} a^{13} - \frac{9651374425}{14317073} a^{12} - \frac{6185563650}{14317073} a^{11} - \frac{25641842100}{14317073} a^{10} - \frac{8905316525}{14317073} a^{9} - \frac{70740393066}{14317073} a^{8} - \frac{1074107475}{14317073} a^{7} - \frac{373028950}{14317073} a^{6} - \frac{129540600}{14317073} a^{5} - \frac{44979375}{14317073} a^{4} - \frac{15592850}{14317073} a^{3} - \frac{5397525}{14317073} a^{2} - \frac{1799175}{14317073} a - \frac{599725}{14317073} \) (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:  \( 15853905121.091976 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 15853905121.091976 \cdot 93}{22\sqrt{159386923550435671074967363509984324121230045171}}\approx 0.157640645261730$ (assuming GRH)

Galois group

$C_{30}$ (as 30T1):

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

Intermediate fields

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{11})^+\), 6.0.8732691.1, \(\Q(\zeta_{11})\), 15.15.10943023107606534329121.1

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

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 $15^{2}$ $30$ R $30$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ $30$ $15^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{6}$ $30$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{5}$ $15^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{6}$ $15^{2}$

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
$3$3.15.20.65$x^{15} + 78 x^{14} + 39 x^{13} + 49 x^{12} + 24 x^{11} + 36 x^{10} + 2 x^{9} + 54 x^{8} + 69 x^{7} + 47 x^{6} + 18 x^{5} + 15 x^{4} + 36 x^{3} + 36 x^{2} + 63 x + 73$$3$$5$$20$$C_{15}$$[2]^{5}$
3.15.20.65$x^{15} + 78 x^{14} + 39 x^{13} + 49 x^{12} + 24 x^{11} + 36 x^{10} + 2 x^{9} + 54 x^{8} + 69 x^{7} + 47 x^{6} + 18 x^{5} + 15 x^{4} + 36 x^{3} + 36 x^{2} + 63 x + 73$$3$$5$$20$$C_{15}$$[2]^{5}$
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) $1$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.99.6t1.a.a$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.99.6t1.a.b$1$ $ 3^{2} \cdot 11 $ 6.0.8732691.1 $C_6$ (as 6T1) $0$ $-1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.99.15t1.a.a$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.30t1.a.a$1$ $ 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.b$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.99.15t1.a.c$1$ $ 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) $0$ $-1$
* 1.99.15t1.a.d$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.30t1.a.d$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.10t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})\) $C_{10}$ (as 10T1) $0$ $-1$
* 1.99.15t1.a.e$1$ $ 3^{2} \cdot 11 $ 15.15.10943023107606534329121.1 $C_{15}$ (as 15T1) $0$ $1$
* 1.99.30t1.a.e$1$ $ 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) $0$ $1$
* 1.99.30t1.a.f$1$ $ 3^{2} \cdot 11 $ 30.0.159386923550435671074967363509984324121230045171.1 $C_{30}$ (as 30T1) $0$ $-1$
* 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) $0$ $-1$
* 1.99.15t1.a.g$1$ $ 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.h$1$ $ 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) $0$ $-1$

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 *.