Properties

Label 30.0.843...984.1
Degree $30$
Signature $[0, 15]$
Discriminant $-8.439\times 10^{50}$
Root discriminant $49.84$
Ramified primes $2, 7, 11$
Class number $2896$ (GRH)
Class group $[2, 2, 2, 362]$ (GRH)
Galois group $C_{30}$ (as 30T1)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 + 45*x^28 + 850*x^26 + 8863*x^24 + 56474*x^22 + 230106*x^20 + 610963*x^18 + 1061766*x^16 + 1203495*x^14 + 883132*x^12 + 414061*x^10 + 121105*x^8 + 21162*x^6 + 2035*x^4 + 90*x^2 + 1)
 
gp: K = bnfinit(x^30 + 45*x^28 + 850*x^26 + 8863*x^24 + 56474*x^22 + 230106*x^20 + 610963*x^18 + 1061766*x^16 + 1203495*x^14 + 883132*x^12 + 414061*x^10 + 121105*x^8 + 21162*x^6 + 2035*x^4 + 90*x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 90, 0, 2035, 0, 21162, 0, 121105, 0, 414061, 0, 883132, 0, 1203495, 0, 1061766, 0, 610963, 0, 230106, 0, 56474, 0, 8863, 0, 850, 0, 45, 0, 1]);
 

\( x^{30} + 45 x^{28} + 850 x^{26} + 8863 x^{24} + 56474 x^{22} + 230106 x^{20} + 610963 x^{18} + 1061766 x^{16} + 1203495 x^{14} + 883132 x^{12} + 414061 x^{10} + 121105 x^{8} + 21162 x^{6} + 2035 x^{4} + 90 x^{2} + 1 \)

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

Invariants

Degree:  $30$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 15]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-843888588175611025887990382060441445435213803945984\)\(\medspace = -\,2^{30}\cdot 7^{20}\cdot 11^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $49.84$
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:  $2, 7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $30$
This field is Galois and abelian over $\Q$.
Conductor:  \(308=2^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{308}(1,·)$, $\chi_{308}(67,·)$, $\chi_{308}(135,·)$, $\chi_{308}(9,·)$, $\chi_{308}(267,·)$, $\chi_{308}(141,·)$, $\chi_{308}(207,·)$, $\chi_{308}(15,·)$, $\chi_{308}(81,·)$, $\chi_{308}(291,·)$, $\chi_{308}(23,·)$, $\chi_{308}(225,·)$, $\chi_{308}(25,·)$, $\chi_{308}(155,·)$, $\chi_{308}(221,·)$, $\chi_{308}(289,·)$, $\chi_{308}(163,·)$, $\chi_{308}(113,·)$, $\chi_{308}(37,·)$, $\chi_{308}(295,·)$, $\chi_{308}(169,·)$, $\chi_{308}(71,·)$, $\chi_{308}(93,·)$, $\chi_{308}(177,·)$, $\chi_{308}(179,·)$, $\chi_{308}(235,·)$, $\chi_{308}(53,·)$, $\chi_{308}(137,·)$, $\chi_{308}(247,·)$, $\chi_{308}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{43} a^{24} + \frac{17}{43} a^{22} - \frac{3}{43} a^{20} + \frac{8}{43} a^{18} + \frac{9}{43} a^{16} - \frac{8}{43} a^{14} + \frac{2}{43} a^{12} - \frac{20}{43} a^{10} + \frac{19}{43} a^{8} + \frac{11}{43} a^{6} + \frac{14}{43} a^{4} - \frac{3}{43} a^{2} + \frac{6}{43}$, $\frac{1}{43} a^{25} + \frac{17}{43} a^{23} - \frac{3}{43} a^{21} + \frac{8}{43} a^{19} + \frac{9}{43} a^{17} - \frac{8}{43} a^{15} + \frac{2}{43} a^{13} - \frac{20}{43} a^{11} + \frac{19}{43} a^{9} + \frac{11}{43} a^{7} + \frac{14}{43} a^{5} - \frac{3}{43} a^{3} + \frac{6}{43} a$, $\frac{1}{43} a^{26} + \frac{9}{43} a^{22} + \frac{16}{43} a^{20} + \frac{2}{43} a^{18} + \frac{11}{43} a^{16} + \frac{9}{43} a^{14} - \frac{11}{43} a^{12} + \frac{15}{43} a^{10} - \frac{11}{43} a^{8} - \frac{1}{43} a^{6} + \frac{17}{43} a^{4} + \frac{14}{43} a^{2} - \frac{16}{43}$, $\frac{1}{43} a^{27} + \frac{9}{43} a^{23} + \frac{16}{43} a^{21} + \frac{2}{43} a^{19} + \frac{11}{43} a^{17} + \frac{9}{43} a^{15} - \frac{11}{43} a^{13} + \frac{15}{43} a^{11} - \frac{11}{43} a^{9} - \frac{1}{43} a^{7} + \frac{17}{43} a^{5} + \frac{14}{43} a^{3} - \frac{16}{43} a$, $\frac{1}{7479011552904015893} a^{28} + \frac{27872587207794719}{7479011552904015893} a^{26} + \frac{18027665817139876}{7479011552904015893} a^{24} + \frac{2570561645828795715}{7479011552904015893} a^{22} + \frac{3334895558423367689}{7479011552904015893} a^{20} + \frac{738691988724068331}{7479011552904015893} a^{18} - \frac{2688289276327114803}{7479011552904015893} a^{16} + \frac{1689182633995547860}{7479011552904015893} a^{14} + \frac{1705267465324045131}{7479011552904015893} a^{12} + \frac{2638757463483278482}{7479011552904015893} a^{10} + \frac{2469847977549552283}{7479011552904015893} a^{8} - \frac{984848737265846731}{7479011552904015893} a^{6} - \frac{900055511948140602}{7479011552904015893} a^{4} - \frac{843257220000486907}{7479011552904015893} a^{2} + \frac{2626829118364790056}{7479011552904015893}$, $\frac{1}{7479011552904015893} a^{29} + \frac{27872587207794719}{7479011552904015893} a^{27} + \frac{18027665817139876}{7479011552904015893} a^{25} + \frac{2570561645828795715}{7479011552904015893} a^{23} + \frac{3334895558423367689}{7479011552904015893} a^{21} + \frac{738691988724068331}{7479011552904015893} a^{19} - \frac{2688289276327114803}{7479011552904015893} a^{17} + \frac{1689182633995547860}{7479011552904015893} a^{15} + \frac{1705267465324045131}{7479011552904015893} a^{13} + \frac{2638757463483278482}{7479011552904015893} a^{11} + \frac{2469847977549552283}{7479011552904015893} a^{9} - \frac{984848737265846731}{7479011552904015893} a^{7} - \frac{900055511948140602}{7479011552904015893} a^{5} - \frac{843257220000486907}{7479011552904015893} a^{3} + \frac{2626829118364790056}{7479011552904015893} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{362}$, which has order $2896$ (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:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( \frac{240426186645}{547969096093} a^{29} + \frac{10760758044601}{547969096093} a^{27} + \frac{201730137486763}{547969096093} a^{25} + \frac{2081105198802305}{547969096093} a^{23} + \frac{13057742718079493}{547969096093} a^{21} + \frac{52003601010595029}{547969096093} a^{19} + \frac{133345846002344315}{547969096093} a^{17} + \frac{219323822981147224}{547969096093} a^{15} + \frac{227195452499811886}{547969096093} a^{13} + \frac{143059303792294046}{547969096093} a^{11} + \frac{50852416356835306}{547969096093} a^{9} + \frac{8349729121173316}{547969096093} a^{7} + \frac{81167359720271}{547969096093} a^{5} - \frac{110889614614127}{547969096093} a^{3} - \frac{6094826512950}{547969096093} a \) (order $4$)
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:  \( 4697581952.048968 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 4697581952.048968 \cdot 2896}{4\sqrt{843888588175611025887990382060441445435213803945984}}\approx 0.109943206724835$ (assuming GRH)

Galois group

$C_{30}$ (as 30T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 30
The 30 conjugacy class representatives for $C_{30}$
Character table for $C_{30}$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.153664.1, 10.0.219503494144.1, 15.15.886528337182930278529.1

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $30$ $15^{2}$ R R ${\href{/LocalNumberField/13.5.0.1}{5} }^{6}$ $15^{2}$ $30$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{6}$ $30$ $15^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $30$ $15^{2}$ $30$

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
2Data not computed
7Data not computed
11Data not computed