Properties

Label 30.0.525...544.1
Degree $30$
Signature $[0, 15]$
Discriminant $-5.251\times 10^{52}$
Root discriminant \(57.19\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{30}$ (as 30T5712)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x + 4)
 
gp: K = bnfinit(y^30 - 4*y + 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 4*x + 4);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 4*x + 4)
 

\( x^{30} - 4x + 4 \) Copy content Toggle raw display

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-52511447916628477265270473131940055074976225290092544\) \(\medspace = -\,2^{30}\cdot 23\cdot 21\!\cdots\!97\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(57.19\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  not computed
Ramified primes:   \(2\), \(23\), \(21263\!\cdots\!44197\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  $\Q(\sqrt{-48905\!\cdots\!16531}$)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{28}+\frac{1}{2}a^{26}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{9}+a^{8}-a^{7}+a^{6}-1$, $\frac{1}{2}a^{23}+\frac{1}{2}a^{22}-\frac{1}{2}a^{20}+\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-a^{9}-a^{8}+a^{6}+a^{5}-a^{4}-a^{3}+a$, $\frac{1}{2}a^{28}+a^{27}+a^{26}+\frac{1}{2}a^{25}+a^{24}+\frac{3}{2}a^{23}+\frac{3}{2}a^{22}+a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+a^{18}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{15}-a^{14}-a^{12}-2a^{11}-2a^{10}-a^{9}-a^{7}-2a^{6}-a^{5}+a^{4}+a^{3}-a+1$, $2a^{29}+2a^{28}+2a^{27}+2a^{26}+\frac{3}{2}a^{25}+\frac{3}{2}a^{24}+2a^{23}+\frac{5}{2}a^{22}+2a^{21}+a^{20}+\frac{1}{2}a^{19}+a^{18}+\frac{3}{2}a^{17}+2a^{16}+2a^{15}+a^{14}-a^{12}+a^{11}+2a^{10}+2a^{9}-a^{7}+a^{4}+a^{2}-9$, $\frac{1}{2}a^{28}+\frac{1}{2}a^{27}+a^{25}+a^{24}+a^{23}+a^{22}+\frac{3}{2}a^{21}+a^{20}+a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{14}+a^{11}+a^{9}+2a^{8}-a^{7}+a^{6}-a^{4}-a^{3}-a^{2}-3$, $a^{28}+a^{27}-a^{25}-\frac{1}{2}a^{24}+a^{23}+\frac{1}{2}a^{22}-a^{21}-a^{20}+a^{18}-2a^{16}-\frac{3}{2}a^{15}+a^{14}+2a^{13}-2a^{11}-a^{10}+2a^{9}+2a^{8}-a^{7}-2a^{6}+3a^{4}+2a^{3}-3a^{2}-3a+2$, $\frac{3}{2}a^{29}+\frac{1}{2}a^{28}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-a^{23}+\frac{3}{2}a^{21}-\frac{3}{2}a^{19}-\frac{3}{2}a^{18}+\frac{1}{2}a^{17}+a^{16}-a^{15}+2a^{13}-2a^{11}-2a^{10}+2a^{9}+3a^{8}-a^{7}-a^{6}+a^{5}-2a^{3}-3a^{2}+4a-1$, $a^{29}+\frac{1}{2}a^{28}+\frac{1}{2}a^{27}+a^{26}+a^{25}+\frac{1}{2}a^{23}+\frac{3}{2}a^{22}+\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{3}{2}a^{18}+\frac{1}{2}a^{17}+a^{14}+a^{13}-a^{11}+a^{10}+2a^{9}-a^{8}-a^{7}+a^{6}+2a^{5}-a^{4}-a^{3}+a-3$, $a^{29}+\frac{3}{2}a^{28}+\frac{3}{2}a^{27}+a^{26}+a^{25}+\frac{1}{2}a^{24}+\frac{1}{2}a^{23}+\frac{3}{2}a^{22}+\frac{3}{2}a^{21}+\frac{3}{2}a^{20}+2a^{19}+a^{18}+a^{17}+\frac{3}{2}a^{16}+a^{13}+a^{12}+2a^{11}+2a^{10}+a^{8}+a^{7}-a^{6}+3a^{2}+a-5$, $\frac{1}{2}a^{29}+a^{26}-\frac{1}{2}a^{25}+\frac{3}{2}a^{24}-\frac{1}{2}a^{23}+\frac{1}{2}a^{22}+a^{21}-\frac{1}{2}a^{20}+2a^{19}-a^{18}+\frac{3}{2}a^{17}-a^{15}+2a^{14}-2a^{13}+2a^{12}-a^{11}+a^{9}-3a^{8}+3a^{7}-3a^{6}+2a^{5}-2a^{3}+4a^{2}-6a+3$, $\frac{1}{2}a^{29}+a^{28}+\frac{1}{2}a^{27}-\frac{1}{2}a^{26}-a^{25}+a^{24}+a^{23}-\frac{1}{2}a^{22}-a^{21}-a^{20}+\frac{3}{2}a^{19}+\frac{3}{2}a^{18}-\frac{3}{2}a^{17}-\frac{3}{2}a^{16}-\frac{1}{2}a^{15}+2a^{14}+2a^{13}-2a^{12}-a^{11}+a^{10}+2a^{9}+a^{8}-4a^{7}-2a^{6}+3a^{5}+2a^{4}-4a^{2}-2a+4$, $\frac{1}{2}a^{28}+\frac{1}{2}a^{27}-\frac{1}{2}a^{25}+a^{23}+a^{22}-\frac{1}{2}a^{21}-\frac{3}{2}a^{20}-a^{19}+a^{18}+a^{17}-\frac{1}{2}a^{16}-2a^{15}-a^{14}+a^{13}+2a^{12}-a^{10}-a^{9}+a^{8}+a^{7}-a^{5}-a+1$, $\frac{3}{2}a^{29}-\frac{1}{2}a^{28}+a^{27}+\frac{3}{2}a^{26}-\frac{1}{2}a^{25}+a^{24}-a^{22}-\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-a^{18}-\frac{1}{2}a^{17}+a^{16}-\frac{7}{2}a^{15}+a^{13}-4a^{12}+2a^{11}-4a^{9}+a^{8}-2a^{7}-a^{6}-a^{4}+2a^{3}-5a^{2}-4$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 552056500614525.9 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{15}\cdot 552056500614525.9 \cdot 1}{2\cdot\sqrt{52511447916628477265270473131940055074976225290092544}}\cr\approx \mathstrut & 1.13116135947245 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 4*x + 4)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^30 - 4*x + 4, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^30 - 4*x + 4);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 4*x + 4);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
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 $\Q$.
sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $19{,}\,{\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ $29{,}\,{\href{/padicField/5.1.0.1}{1} }$ $27{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ $22{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ $16{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ $30$ ${\href{/padicField/19.13.0.1}{13} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ R $29{,}\,{\href{/padicField/29.1.0.1}{1} }$ $22{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.11.0.1}{11} }{,}\,{\href{/padicField/41.5.0.1}{5} }$ ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ $29{,}\,{\href{/padicField/47.1.0.1}{1} }$ $21{,}\,{\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ $15{,}\,{\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $30$$30$$1$$30$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
23.2.1.1$x^{2} + 115$$2$$1$$1$$C_2$$[\ ]_{2}$
23.5.0.1$x^{5} + 3 x + 18$$1$$5$$0$$C_5$$[\ ]^{5}$
23.6.0.1$x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
23.15.0.1$x^{15} + 2 x^{6} + 8 x^{5} + 15 x^{4} + 9 x^{3} + 7 x^{2} + 18 x + 18$$1$$15$$0$$C_{15}$$[\ ]^{15}$
\(212\!\cdots\!197\) Copy content Toggle raw display $\Q_{21\!\cdots\!97}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $19$$1$$19$$0$$C_{19}$$[\ ]^{19}$