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Properties

Label 30.2.698...512.1
Degree $30$
Signature $[2, 14]$
Discriminant $6.981\times 10^{46}$
Root discriminant \(36.43\)
Ramified primes $2,239$
Class number not computed
Class group not computed
Galois group $D_{30}$ (as 30T14)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768)
 
gp: K = bnfinit(y^30 - 16*y^28 + 152*y^26 - 960*y^24 + 4656*y^22 - 18272*y^20 + 59264*y^18 - 161280*y^16 + 368896*y^14 - 701952*y^12 + 1121280*y^10 - 1376256*y^8 + 1417216*y^6 - 884736*y^4 + 524288*y^2 - 32768, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768)
 

\( x^{30} - 16 x^{28} + 152 x^{26} - 960 x^{24} + 4656 x^{22} - 18272 x^{20} + 59264 x^{18} - 161280 x^{16} + \cdots - 32768 \) 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:   \(69810446891843341455650582518642209772915392512\) \(\medspace = 2^{45}\cdot 239^{14}\) 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:  \(36.43\)
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:  $2^{3/2}239^{1/2}\approx 43.726422218150894$
Ramified primes:   \(2\), \(239\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{53248}a^{24}-\frac{3}{13312}a^{22}-\frac{3}{3328}a^{18}-\frac{1}{1664}a^{16}-\frac{3}{1664}a^{14}+\frac{1}{416}a^{12}-\frac{1}{208}a^{10}+\frac{3}{208}a^{8}-\frac{3}{52}a^{6}-\frac{3}{52}a^{4}+\frac{1}{13}a^{2}+\frac{4}{13}$, $\frac{1}{53248}a^{25}-\frac{3}{13312}a^{23}-\frac{3}{3328}a^{19}-\frac{1}{1664}a^{17}-\frac{3}{1664}a^{15}+\frac{1}{416}a^{13}-\frac{1}{208}a^{11}+\frac{3}{208}a^{9}-\frac{3}{52}a^{7}-\frac{3}{52}a^{5}+\frac{1}{13}a^{3}+\frac{4}{13}a$, $\frac{1}{106496}a^{26}+\frac{3}{26624}a^{22}-\frac{3}{6656}a^{20}+\frac{1}{6656}a^{18}-\frac{1}{1664}a^{16}-\frac{3}{1664}a^{14}-\frac{3}{832}a^{12}+\frac{1}{104}a^{10}-\frac{1}{208}a^{8}-\frac{3}{52}a^{4}+\frac{3}{26}a^{2}-\frac{2}{13}$, $\frac{1}{106496}a^{27}+\frac{3}{26624}a^{23}-\frac{3}{6656}a^{21}+\frac{1}{6656}a^{19}-\frac{1}{1664}a^{17}-\frac{3}{1664}a^{15}-\frac{3}{832}a^{13}+\frac{1}{104}a^{11}-\frac{1}{208}a^{9}-\frac{3}{52}a^{5}+\frac{3}{26}a^{3}-\frac{2}{13}a$, $\frac{1}{86189756563456}a^{28}-\frac{8108245}{3314990637056}a^{26}-\frac{4714025}{1346714946304}a^{24}+\frac{880718637}{10773719570432}a^{22}-\frac{1173121429}{5386859785216}a^{20}+\frac{6714797}{24048481184}a^{18}-\frac{26681021}{336678736576}a^{16}+\frac{1772222793}{673357473152}a^{14}+\frac{46550033}{6474591088}a^{12}-\frac{153683449}{12024240592}a^{10}+\frac{148716655}{10521210518}a^{8}+\frac{682269079}{21042421036}a^{6}-\frac{402203244}{5260605259}a^{4}+\frac{418113485}{5260605259}a^{2}+\frac{2115234932}{5260605259}$, $\frac{1}{86189756563456}a^{29}-\frac{8108245}{3314990637056}a^{27}-\frac{4714025}{1346714946304}a^{25}+\frac{880718637}{10773719570432}a^{23}-\frac{1173121429}{5386859785216}a^{21}+\frac{6714797}{24048481184}a^{19}-\frac{26681021}{336678736576}a^{17}+\frac{1772222793}{673357473152}a^{15}+\frac{46550033}{6474591088}a^{13}-\frac{153683449}{12024240592}a^{11}+\frac{148716655}{10521210518}a^{9}+\frac{682269079}{21042421036}a^{7}-\frac{402203244}{5260605259}a^{5}+\frac{418113485}{5260605259}a^{3}+\frac{2115234932}{5260605259}a$ 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

not computed

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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^30 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768)
 
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 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768, 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 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768);
 
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 - 16*x^28 + 152*x^26 - 960*x^24 + 4656*x^22 - 18272*x^20 + 59264*x^18 - 161280*x^16 + 368896*x^14 - 701952*x^12 + 1121280*x^10 - 1376256*x^8 + 1417216*x^6 - 884736*x^4 + 524288*x^2 - 32768);
 
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

$D_{30}$ (as 30T14):

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 solvable group of order 60
The 18 conjugacy class representatives for $D_{30}$
Character table for $D_{30}$

Intermediate fields

\(\Q(\sqrt{2}) \), 3.1.239.1, 5.1.57121.1, 6.2.29245952.1, 10.2.106915713548288.1, 15.1.44543599279432079.1

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

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]
 

Sibling fields

Degree 30 sibling: deg 30
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{3}$ $30$ ${\href{/padicField/7.2.0.1}{2} }^{14}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ $30$ ${\href{/padicField/13.2.0.1}{2} }^{15}$ ${\href{/padicField/17.3.0.1}{3} }^{10}$ ${\href{/padicField/19.2.0.1}{2} }^{15}$ ${\href{/padicField/23.2.0.1}{2} }^{14}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $30$ $15^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{15}$ ${\href{/padicField/41.2.0.1}{2} }^{14}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{15}$ ${\href{/padicField/47.2.0.1}{2} }^{14}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{15}$ ${\href{/padicField/59.2.0.1}{2} }^{15}$

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$$2$$15$$45$
\(239\) Copy content Toggle raw display $\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{239}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.1912.2t1.b.a$1$ $ 2^{3} \cdot 239 $ \(\Q(\sqrt{-478}) \) $C_2$ (as 2T1) $1$ $-1$
1.239.2t1.a.a$1$ $ 239 $ \(\Q(\sqrt{-239}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
* 2.15296.6t3.a.a$2$ $ 2^{6} \cdot 239 $ 6.0.6989782528.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.239.3t2.a.a$2$ $ 239 $ 3.1.239.1 $S_3$ (as 3T2) $1$ $0$
* 2.239.5t2.a.a$2$ $ 239 $ 5.1.57121.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.239.5t2.a.b$2$ $ 239 $ 5.1.57121.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.15296.10t3.a.a$2$ $ 2^{6} \cdot 239 $ 10.0.25552855538040832.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.15296.10t3.a.b$2$ $ 2^{6} \cdot 239 $ 10.0.25552855538040832.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.239.15t2.a.b$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.15296.30t14.a.b$2$ $ 2^{6} \cdot 239 $ 30.2.69810446891843341455650582518642209772915392512.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.d$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.15296.30t14.a.a$2$ $ 2^{6} \cdot 239 $ 30.2.69810446891843341455650582518642209772915392512.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.c$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.15296.30t14.a.d$2$ $ 2^{6} \cdot 239 $ 30.2.69810446891843341455650582518642209772915392512.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.239.15t2.a.a$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.15296.30t14.a.c$2$ $ 2^{6} \cdot 239 $ 30.2.69810446891843341455650582518642209772915392512.1 $D_{30}$ (as 30T14) $1$ $0$

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 *.