# Properties

 Label 30.2.528...381.1 Degree $30$ Signature $[2, 14]$ Discriminant $5.287\times 10^{56}$ Root discriminant $77.76$ Ramified primes see page Class number $1$ (GRH) Class group trivial (GRH) Galois group $S_{30}$ (as 30T5712)

# Learn more

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x + 1)

gp: K = bnfinit(x^30 - 3*x + 1, 1)

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

$$x^{30} - 3 x + 1$$

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: $$528663808937910174564399958795650464488012557052441857381$$$$\medspace = 3^{30}\cdot 7\cdot 83\cdot 547\cdot 8079388286476418501045691928532147267$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $77.76$ 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: $3, 7, 83, 547, 8079388286476418501045691928532147267$ 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 $$a$$)

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

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: $15$ 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: $$27174736235759336$$ (assuming GRH) sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{14}\cdot 27174736235759336 \cdot 1}{2\sqrt{528663808937910174564399958795650464488012557052441857381}}\approx 0.353283781011431$ (assuming GRH)

## 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 $29{,}\,{\href{/padicField/5.1.0.1}{1} }$ R $16{,}\,{\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.10.0.1}{10} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ $30$ $16{,}\,{\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ $18{,}\,{\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.7.0.1}{7} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.11.0.1}{11} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ $16{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ $26{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.6.0.1}{6} }$ $25{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.10.0.1}{10} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ $30$

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$3.6.6.2$x^{6} + 6 x^{4} + 6 x^{3} + 18$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2} 3.12.12.13x^{12} + 42 x^{11} - 57 x^{10} - 36 x^{9} - 117 x^{8} - 54 x^{7} - 18 x^{6} - 27 x^{5} + 54 x^{4} + 108 x^{3} - 81 x^{2} + 81 x + 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
3.12.12.18$x^{12} + 42 x^{11} - 48 x^{10} - 114 x^{9} - 99 x^{8} - 54 x^{7} - 90 x^{6} - 108 x^{5} + 27 x^{4} - 27 x^{3} + 81 x^{2} + 81 x - 81$$3$$4$$1212T46[3/2, 3/2]_{2}^{4} 77.2.1.2x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.0.1$x^{4} + x^{2} - 3 x + 5$$1$$4$$0$$C_4$$[\ ]^{4} 7.6.0.1x^{6} + 3 x^{2} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7} 7.11.0.1x^{11} - 2 x + 4$$1$$11$$0$$C_{11}$$[\ ]^{11}$
83Data not computed
547Data not computed
8079388286476418501045691928532147267Data not computed