Properties

Label 30.2.528...381.1
Degree $30$
Signature $[2, 14]$
Discriminant $5.288\times 10^{56}$
Root discriminant \(77.76\)
Ramified primes see page
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_{30}$ (as 30T5712)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x - 2)
 
gp: K = bnfinit(y^30 - 3*y - 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 3*x - 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 3*x - 2)
 

\( x^{30} - 3x - 2 \) 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:  $[2, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(528774345897976432375444495795650464488012557052441857381\) \(\medspace = 3^{30}\cdot 31\cdot 1769381441963\cdot 46821958249121311829160660473\) 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:  \(77.76\)
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:   \(3\), \(31\), \(1769381441963\), \(46821958249121311829160660473\) 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:  $\Q(\sqrt{25682\!\cdots\!83469}$)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

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

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

Monogenic:  Yes
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:  $15$
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:   $a^{29}+a^{27}+a^{25}+a^{23}+a^{21}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+2a^{8}+a^{7}+2a^{6}+a^{5}+2a^{4}+a^{3}+2a^{2}+a-1$, $a^{29}+a^{28}+a^{27}+a^{26}+a^{25}+a^{24}+a^{23}+a^{22}+a^{21}+a^{20}+a^{19}+a^{18}+a^{17}+a^{16}+a^{15}+a^{14}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+2a+1$, $3a^{29}-2a^{28}-a^{25}-2a^{24}-2a^{21}+a^{20}+2a^{19}-a^{18}+2a^{17}+3a^{16}+2a^{13}-a^{12}-4a^{11}-2a^{9}-5a^{8}+a^{6}-a^{5}+2a^{4}+6a^{3}+2a^{2}+a-3$, $2a^{29}-2a^{28}+2a^{27}-a^{26}+a^{24}-a^{23}+a^{22}-a^{20}+2a^{19}-3a^{18}+3a^{17}-2a^{16}+a^{15}+a^{14}-2a^{13}+3a^{12}-3a^{11}+3a^{10}-a^{9}+a^{8}+2a^{7}-a^{6}+2a^{5}+3a^{2}-2a-1$, $a^{27}-a^{26}-a^{25}+a^{24}+2a^{23}-2a^{22}-2a^{21}+2a^{20}+2a^{19}-2a^{18}-a^{17}+a^{16}+a^{13}-a^{12}-2a^{11}+2a^{10}+2a^{9}-2a^{8}-2a^{7}+2a^{6}+a^{5}-a^{4}-a^{3}+a^{2}-1$, $a^{29}+a^{26}-a^{24}-a^{20}+a^{18}-a^{17}+2a^{15}+a^{14}+a^{12}+a^{11}-2a^{9}+a^{7}-2a^{6}-a^{5}+a^{4}-a^{2}+a+1$, $3a^{29}+4a^{28}+3a^{27}-a^{25}-a^{24}+a^{23}+5a^{22}+6a^{21}+4a^{20}+a^{19}-3a^{18}-4a^{17}+a^{16}+6a^{15}+10a^{14}+11a^{13}+4a^{12}-3a^{11}-6a^{10}-5a^{9}+3a^{8}+13a^{7}+15a^{6}+12a^{5}+4a^{4}-6a^{3}-6a^{2}+a+1$, $2a^{29}+a^{28}-2a^{26}-2a^{25}+2a^{23}+2a^{22}+a^{21}-a^{20}-3a^{19}-3a^{18}-a^{17}+3a^{16}+6a^{15}+5a^{14}-5a^{12}-7a^{11}-5a^{10}+6a^{8}+9a^{7}+6a^{6}-2a^{5}-8a^{4}-9a^{3}-5a^{2}+2a+1$, $a^{29}-a^{28}+2a^{26}-a^{25}-2a^{24}+2a^{22}-a^{21}-2a^{20}+3a^{19}+2a^{18}-2a^{17}-2a^{16}+2a^{15}+a^{14}-4a^{13}-a^{12}+3a^{11}+a^{10}-a^{9}+a^{8}+3a^{7}-2a^{6}-3a^{5}+a^{4}-a^{3}-2a^{2}+a+1$, $2a^{29}-2a^{28}+2a^{27}-a^{26}+2a^{25}-3a^{24}+a^{23}+2a^{21}+a^{19}-2a^{18}-a^{17}+a^{16}+a^{15}-a^{13}-a^{12}-3a^{11}+3a^{10}+2a^{9}+2a^{8}-3a^{7}+2a^{6}-3a^{5}+3a^{4}+2a^{3}+3a^{2}-7a-7$, $a^{29}-2a^{27}+2a^{26}+a^{24}-2a^{22}+2a^{21}-a^{20}+a^{19}-a^{18}-3a^{17}+2a^{16}-a^{15}+3a^{14}-2a^{13}-2a^{12}+3a^{11}+4a^{9}-4a^{8}-a^{7}+2a^{6}+2a^{4}-7a^{3}-a^{2}+2a-1$, $19a^{29}-12a^{28}+8a^{27}-5a^{26}+2a^{25}-a^{24}+a^{22}-a^{21}+a^{20}+a^{17}-2a^{16}+2a^{15}-2a^{14}+3a^{13}-2a^{12}+a^{11}-a^{10}+2a^{8}-2a^{7}+2a^{6}-3a^{5}+2a^{4}-2a^{3}-a-57$, $56a^{29}-37a^{28}+26a^{27}-17a^{26}+11a^{25}-7a^{24}+4a^{23}-3a^{22}+a^{21}-2a^{20}+2a^{19}-2a^{18}+a^{17}+a^{15}+2a^{14}+a^{11}-a^{10}-a^{9}-a^{8}-2a^{7}-3a^{5}-2a^{4}+a^{3}+2a-167$, $3a^{29}+a^{28}-2a^{27}-3a^{26}-a^{25}+2a^{24}+4a^{23}-3a^{21}-4a^{20}-a^{19}+4a^{18}+4a^{17}-5a^{15}-5a^{14}+6a^{12}+6a^{11}-a^{10}-7a^{9}-6a^{8}+a^{7}+9a^{6}+8a^{5}-2a^{4}-9a^{3}-8a^{2}+3a+3$, $7a^{29}-7a^{28}+7a^{27}-6a^{26}+5a^{25}-4a^{24}+a^{23}+a^{22}-4a^{21}+7a^{20}-8a^{19}+9a^{18}-9a^{17}+6a^{16}-5a^{15}+2a^{14}+a^{13}-3a^{12}+7a^{11}-10a^{10}+11a^{9}-13a^{8}+9a^{7}-6a^{6}+2a^{5}+5a^{4}-8a^{3}+9a^{2}-12a-13$ 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:  \( 96404046034030270 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{14}\cdot 96404046034030270 \cdot 1}{2\cdot\sqrt{528774345897976432375444495795650464488012557052441857381}}\cr\approx \mathstrut & 1.25316491030658 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 3*x - 2)
 
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 - 3*x - 2, 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 - 3*x - 2);
 
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 - 3*x - 2);
 
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 $28{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ R $22{,}\,{\href{/padicField/5.8.0.1}{8} }$ ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.11.0.1}{11} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ $29{,}\,{\href{/padicField/11.1.0.1}{1} }$ $22{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ $18{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $29{,}\,{\href{/padicField/19.1.0.1}{1} }$ $18{,}\,{\href{/padicField/23.12.0.1}{12} }$ $30$ R ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.5.0.1}{5} }$ $28{,}\,{\href{/padicField/41.2.0.1}{2} }$ $26{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.8.0.1}{8} }^{2}{,}\,{\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ $15{,}\,{\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ $30$

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

# 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
\(3\) Copy content Toggle raw display 3.6.6.1$x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$$3$$2$$6$$C_3^2:C_4$$[3/2, 3/2]_{2}^{2}$
3.12.12.18$x^{12} - 36 x^{11} + 438 x^{10} + 120 x^{9} - 4563 x^{8} + 1188 x^{7} + 22410 x^{6} + 10692 x^{5} - 17658 x^{4} - 3780 x^{3} + 6804 x^{2} - 1296 x + 81$$3$$4$$12$12T46$[3/2, 3/2]_{2}^{4}$
3.12.12.10$x^{12} - 24 x^{11} + 306 x^{10} - 2004 x^{9} + 7236 x^{8} - 4374 x^{7} - 1458 x^{6} + 5832 x^{5} - 1836 x^{3} + 324 x^{2} + 486 x + 81$$3$$4$$12$12T173$[3/2, 3/2, 3/2, 3/2]_{2}^{4}$
\(31\) Copy content Toggle raw display 31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $28$$1$$28$$0$$C_{28}$$[\ ]^{28}$
\(1769381441963\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $4$$1$$4$$0$$C_4$$[\ ]^{4}$
Deg $5$$1$$5$$0$$C_5$$[\ ]^{5}$
Deg $6$$1$$6$$0$$C_6$$[\ ]^{6}$
Deg $11$$1$$11$$0$$C_{11}$$[\ ]^{11}$
\(468\!\cdots\!473\) Copy content Toggle raw display Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $8$$1$$8$$0$$C_8$$[\ ]^{8}$
Deg $20$$1$$20$$0$20T1$[\ ]^{20}$