# Properties

 Label 30.0.426...291.1 Degree $30$ Signature $[0, 15]$ Discriminant $-4.261\times 10^{63}$ Root discriminant $132.12$ Ramified primes see page Class number not computed Class group not computed Galois group $S_{30}$ (as 30T5712)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x + 5)

gp: K = bnfinit(x^30 - 3*x + 5, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);

$$x^{30} - 3 x + 5$$

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: $$-42\!\cdots\!291$$$$\medspace = -\,3^{29}\cdot 71\cdot 467\cdot 51628578737\cdot 9899468832468151\cdot 3663798440144491003$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $132.12$ 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: $3, 71, 467, 51628578737, 9899468832468151, 3663798440144491003$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{3} a^{29} - \frac{1}{3} a^{28} + \frac{1}{3} a^{27} - \frac{1}{3} a^{26} + \frac{1}{3} a^{25} - \frac{1}{3} a^{24} + \frac{1}{3} a^{23} - \frac{1}{3} a^{22} + \frac{1}{3} a^{21} - \frac{1}{3} a^{20} + \frac{1}{3} a^{19} - \frac{1}{3} a^{18} + \frac{1}{3} a^{17} - \frac{1}{3} a^{16} + \frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

not computed

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: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: not computed sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: not computed sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s)$ not computed

## Galois group

$S_{30}$ (as 30T5712):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 265252859812191058636308480000000 The 5604 conjugacy class representatives for $S_{30}$ are not computed Character table for $S_{30}$ is not computed

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## 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 ${\href{/padicField/5.14.0.1}{14} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ $18{,}\,{\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ $26{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ $18{,}\,{\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.7.0.1}{7} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.5.0.1}{5} }^{6}$ $16{,}\,{\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ $19{,}\,{\href{/padicField/37.11.0.1}{11} }$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ $16{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ $15{,}\,{\href{/padicField/47.10.0.1}{10} }{,}\,{\href{/padicField/47.5.0.1}{5} }$ ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $30$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0Trivial[\ ] 3.2.1.1x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.4.4$x^{3} + 3 x^{2} + 3$$3$$1$$4$$S_3$$[2]^{2} 3.12.12.2x^{12} + 78 x^{10} - 75 x^{9} - 36 x^{8} + 36 x^{7} + 81 x^{6} - 81 x^{5} - 81 x^{4} + 81 x^{3} + 81 x^{2} + 81 x - 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
3.12.12.22$x^{12} + 18 x^{11} + 21 x^{10} - 69 x^{9} - 81 x^{8} + 72 x^{7} - 90 x^{6} - 108 x^{5} + 54 x^{4} - 108 x^{3} - 81$$3$$4$$1212T119[3/2, 3/2, 3/2]_{2}^{4} 71$$\Q_{71}$$x + 2$$1$$1$$0$Trivial$[\ ]$
71.2.1.2$x^{2} + 142$$2$$1$$1$$C_2$$[\ ]_{2} 71.7.0.1x^{7} - x + 6$$1$$7$$0$$C_7$$[\ ]^{7}$
71.10.0.1$x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10} 71.10.0.1x^{10} - x + 22$$1$$10$$0$$C_{10}$$[\ ]^{10}$
467Data not computed
51628578737Data not computed
9899468832468151Data not computed
3663798440144491003Data not computed