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Properties

Label 30.2.305...688.1
Degree $30$
Signature $[2, 14]$
Discriminant $3.057\times 10^{49}$
Root discriminant \(44.62\)
Ramified primes $2,3,239$
Class number not computed
Class group not computed
Galois group $D_{30}$ (as 30T14)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907)
 
gp: K = bnfinit(y^30 - 24*y^28 + 342*y^26 - 3240*y^24 + 23571*y^22 - 138753*y^20 + 675054*y^18 - 2755620*y^16 + 9454401*y^14 - 26985393*y^12 + 64658655*y^10 - 119042784*y^8 + 183878586*y^6 - 172186884*y^4 + 153055008*y^2 - 14348907, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^30 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^30 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907)
 

\( x^{30} - 24 x^{28} + 342 x^{26} - 3240 x^{24} + 23571 x^{22} - 138753 x^{20} + 675054 x^{18} + \cdots - 14348907 \) 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:   \(30569568178695653232311243684564905832093935730688\) \(\medspace = 2^{30}\cdot 3^{15}\cdot 239^{14}\) 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:  \(44.62\)
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:  $2\cdot 3^{1/2}239^{1/2}\approx 53.55371135598354$
Ramified primes:   \(2\), \(3\), \(239\) 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:  \(\Q(\sqrt{3}) \)
$\card{ \Aut(K/\Q) }$:  $2$
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 \(a\))

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{9}a^{4}$, $\frac{1}{9}a^{5}$, $\frac{1}{27}a^{6}$, $\frac{1}{27}a^{7}$, $\frac{1}{81}a^{8}$, $\frac{1}{81}a^{9}$, $\frac{1}{243}a^{10}$, $\frac{1}{243}a^{11}$, $\frac{1}{729}a^{12}$, $\frac{1}{729}a^{13}$, $\frac{1}{2187}a^{14}$, $\frac{1}{2187}a^{15}$, $\frac{1}{6561}a^{16}$, $\frac{1}{6561}a^{17}$, $\frac{1}{19683}a^{18}$, $\frac{1}{19683}a^{19}$, $\frac{1}{59049}a^{20}$, $\frac{1}{59049}a^{21}$, $\frac{1}{177147}a^{22}$, $\frac{1}{177147}a^{23}$, $\frac{1}{6908733}a^{24}-\frac{2}{767637}a^{22}-\frac{2}{85293}a^{18}-\frac{2}{85293}a^{16}-\frac{1}{9477}a^{14}+\frac{2}{9477}a^{12}-\frac{2}{3159}a^{10}+\frac{1}{351}a^{8}-\frac{2}{117}a^{6}-\frac{1}{39}a^{4}+\frac{2}{39}a^{2}+\frac{4}{13}$, $\frac{1}{6908733}a^{25}-\frac{2}{767637}a^{23}-\frac{2}{85293}a^{19}-\frac{2}{85293}a^{17}-\frac{1}{9477}a^{15}+\frac{2}{9477}a^{13}-\frac{2}{3159}a^{11}+\frac{1}{351}a^{9}-\frac{2}{117}a^{7}-\frac{1}{39}a^{5}+\frac{2}{39}a^{3}+\frac{4}{13}a$, $\frac{1}{20726199}a^{26}+\frac{1}{767637}a^{22}-\frac{2}{255879}a^{20}+\frac{1}{255879}a^{18}-\frac{2}{85293}a^{16}-\frac{1}{9477}a^{14}-\frac{1}{3159}a^{12}+\frac{4}{3159}a^{10}-\frac{1}{1053}a^{8}-\frac{1}{39}a^{4}+\frac{1}{13}a^{2}-\frac{2}{13}$, $\frac{1}{20726199}a^{27}+\frac{1}{767637}a^{23}-\frac{2}{255879}a^{21}+\frac{1}{255879}a^{19}-\frac{2}{85293}a^{17}-\frac{1}{9477}a^{15}-\frac{1}{3159}a^{13}+\frac{4}{3159}a^{11}-\frac{1}{1053}a^{9}-\frac{1}{39}a^{5}+\frac{1}{13}a^{3}-\frac{2}{13}a$, $\frac{1}{25\!\cdots\!71}a^{28}-\frac{8108245}{645161842949589}a^{26}-\frac{75424400}{27\!\cdots\!19}a^{24}+\frac{293572879}{310633479938691}a^{22}-\frac{1173121429}{310633479938691}a^{20}+\frac{107436752}{14792070473271}a^{18}-\frac{106724084}{34514831104299}a^{16}+\frac{590740931}{3834981233811}a^{14}+\frac{186200132}{294998556447}a^{12}-\frac{307366898}{182618153991}a^{10}+\frac{1189733240}{426109025979}a^{8}+\frac{1364538158}{142036341993}a^{6}-\frac{536270992}{15781815777}a^{4}+\frac{836226970}{15781815777}a^{2}+\frac{2115234932}{5260605259}$, $\frac{1}{25\!\cdots\!71}a^{29}-\frac{8108245}{645161842949589}a^{27}-\frac{75424400}{27\!\cdots\!19}a^{25}+\frac{293572879}{310633479938691}a^{23}-\frac{1173121429}{310633479938691}a^{21}+\frac{107436752}{14792070473271}a^{19}-\frac{106724084}{34514831104299}a^{17}+\frac{590740931}{3834981233811}a^{15}+\frac{186200132}{294998556447}a^{13}-\frac{307366898}{182618153991}a^{11}+\frac{1189733240}{426109025979}a^{9}+\frac{1364538158}{142036341993}a^{7}-\frac{536270992}{15781815777}a^{5}+\frac{836226970}{15781815777}a^{3}+\frac{2115234932}{5260605259}a$ 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 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907)
 
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 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907, 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 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907);
 
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 - 24*x^28 + 342*x^26 - 3240*x^24 + 23571*x^22 - 138753*x^20 + 675054*x^18 - 2755620*x^16 + 9454401*x^14 - 26985393*x^12 + 64658655*x^10 - 119042784*x^8 + 183878586*x^6 - 172186884*x^4 + 153055008*x^2 - 14348907);
 
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

\(\Q(\sqrt{3}) \), 3.1.239.1, 5.1.57121.1, 6.2.98705088.2, 10.2.811891199757312.1, 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 R $30$ ${\href{/padicField/7.2.0.1}{2} }^{15}$ $15^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{14}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.6.0.1}{6} }^{5}$ ${\href{/padicField/19.2.0.1}{2} }^{15}$ ${\href{/padicField/23.2.0.1}{2} }^{14}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ $30$ $30$ ${\href{/padicField/37.2.0.1}{2} }^{14}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{15}$ ${\href{/padicField/43.2.0.1}{2} }^{15}$ ${\href{/padicField/47.2.0.1}{2} }^{14}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{15}$ ${\href{/padicField/59.2.0.1}{2} }^{14}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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$$30$
\(3\) Copy content Toggle raw display 3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.1$x^{10} + 162 x^{2} - 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
\(239\) 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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2868.2t1.a.a$1$ $ 2^{2} \cdot 3 \cdot 239 $ \(\Q(\sqrt{-717}) \) $C_2$ (as 2T1) $1$ $-1$
1.239.2t1.a.a$1$ $ 239 $ \(\Q(\sqrt{-239}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.12.2t1.a.a$1$ $ 2^{2} \cdot 3 $ \(\Q(\sqrt{3}) \) $C_2$ (as 2T1) $1$ $1$
* 2.34416.6t3.a.a$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 6.0.23590516032.3 $D_{6}$ (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) $1$ $0$
* 2.34416.10t3.a.a$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 10.0.194041996741997568.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.34416.10t3.a.b$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 10.0.194041996741997568.1 $D_{10}$ (as 10T3) $1$ $0$
* 2.239.15t2.a.b$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.239.15t2.a.a$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.239.15t2.a.d$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.239.15t2.a.c$2$ $ 239 $ 15.1.44543599279432079.1 $D_{15}$ (as 15T2) $1$ $0$
* 2.34416.30t14.a.a$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 30.2.30569568178695653232311243684564905832093935730688.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.34416.30t14.a.d$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 30.2.30569568178695653232311243684564905832093935730688.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.34416.30t14.a.b$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 30.2.30569568178695653232311243684564905832093935730688.1 $D_{30}$ (as 30T14) $1$ $0$
* 2.34416.30t14.a.c$2$ $ 2^{4} \cdot 3^{2} \cdot 239 $ 30.2.30569568178695653232311243684564905832093935730688.1 $D_{30}$ (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 *.