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Properties

Label 30.0.104...371.1
Degree $30$
Signature $[0, 15]$
Discriminant $-1.046\times 10^{45}$
Root discriminant $31.67$
Ramified primes $7, 11$
Class number $16$ (GRH)
Class group $[2, 2, 2, 2]$ (GRH)
Galois group $C_{30}$ (as 30T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1)
 
gp: K = bnfinit(x^30 - x^29 + 3*x^28 - 4*x^27 + 9*x^26 - 14*x^25 + 28*x^24 - 47*x^23 + 89*x^22 - 155*x^21 + 286*x^20 + 132*x^19 + 285*x^18 + 265*x^17 + 437*x^16 + 378*x^15 + 761*x^14 + 432*x^13 + 1468*x^12 + 157*x^11 + 3211*x^10 - 1429*x^9 + 636*x^8 - 283*x^7 + 126*x^6 - 56*x^5 + 25*x^4 - 11*x^3 + 5*x^2 - 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 5, -11, 25, -56, 126, -283, 636, -1429, 3211, 157, 1468, 432, 761, 378, 437, 265, 285, 132, 286, -155, 89, -47, 28, -14, 9, -4, 3, -1, 1]);
 

\( x^{30} - x^{29} + 3 x^{28} - 4 x^{27} + 9 x^{26} - 14 x^{25} + 28 x^{24} - 47 x^{23} + 89 x^{22} - 155 x^{21} + 286 x^{20} + 132 x^{19} + 285 x^{18} + 265 x^{17} + 437 x^{16} + 378 x^{15} + 761 x^{14} + 432 x^{13} + 1468 x^{12} + 157 x^{11} + 3211 x^{10} - 1429 x^{9} + 636 x^{8} - 283 x^{7} + 126 x^{6} - 56 x^{5} + 25 x^{4} - 11 x^{3} + 5 x^{2} - 2 x + 1 \)

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:  \(-1046076147688308987260717152173116396995512371\)\(\medspace = -\,7^{20}\cdot 11^{27}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $31.67$
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:  $7, 11$
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:  \(77=7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{77}(64,·)$, $\chi_{77}(1,·)$, $\chi_{77}(2,·)$, $\chi_{77}(67,·)$, $\chi_{77}(4,·)$, $\chi_{77}(65,·)$, $\chi_{77}(8,·)$, $\chi_{77}(9,·)$, $\chi_{77}(74,·)$, $\chi_{77}(15,·)$, $\chi_{77}(16,·)$, $\chi_{77}(18,·)$, $\chi_{77}(23,·)$, $\chi_{77}(25,·)$, $\chi_{77}(71,·)$, $\chi_{77}(29,·)$, $\chi_{77}(30,·)$, $\chi_{77}(32,·)$, $\chi_{77}(36,·)$, $\chi_{77}(37,·)$, $\chi_{77}(39,·)$, $\chi_{77}(43,·)$, $\chi_{77}(46,·)$, $\chi_{77}(72,·)$, $\chi_{77}(50,·)$, $\chi_{77}(51,·)$, $\chi_{77}(53,·)$, $\chi_{77}(57,·)$, $\chi_{77}(58,·)$, $\chi_{77}(60,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{696851} a^{21} - \frac{332692}{696851} a^{20} + \frac{335130}{696851} a^{19} - \frac{303662}{696851} a^{18} - \frac{55621}{696851} a^{17} - \frac{216573}{696851} a^{16} - \frac{198331}{696851} a^{15} - \frac{290436}{696851} a^{14} - \frac{322799}{696851} a^{13} + \frac{240447}{696851} a^{12} + \frac{217221}{696851} a^{11} + \frac{319182}{696851} a^{10} + \frac{341691}{696851} a^{9} + \frac{138309}{696851} a^{8} + \frac{167404}{696851} a^{7} - \frac{245946}{696851} a^{6} + \frac{22212}{696851} a^{5} - \frac{346700}{696851} a^{4} + \frac{145178}{696851} a^{3} - \frac{119515}{696851} a^{2} + \frac{63171}{696851} a - \frac{157023}{696851}$, $\frac{1}{696851} a^{22} - \frac{318543}{696851} a^{11} - \frac{163850}{696851}$, $\frac{1}{696851} a^{23} - \frac{318543}{696851} a^{12} - \frac{163850}{696851} a$, $\frac{1}{696851} a^{24} - \frac{318543}{696851} a^{13} - \frac{163850}{696851} a^{2}$, $\frac{1}{696851} a^{25} - \frac{318543}{696851} a^{14} - \frac{163850}{696851} a^{3}$, $\frac{1}{696851} a^{26} - \frac{318543}{696851} a^{15} - \frac{163850}{696851} a^{4}$, $\frac{1}{696851} a^{27} - \frac{318543}{696851} a^{16} - \frac{163850}{696851} a^{5}$, $\frac{1}{696851} a^{28} - \frac{318543}{696851} a^{17} - \frac{163850}{696851} a^{6}$, $\frac{1}{696851} a^{29} - \frac{318543}{696851} a^{18} - \frac{163850}{696851} a^{7}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}$, which has order $16$ (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:  \( -\frac{310114}{696851} a^{29} + \frac{310114}{696851} a^{28} - \frac{930342}{696851} a^{27} + \frac{1240456}{696851} a^{26} - \frac{2791026}{696851} a^{25} + \frac{4341596}{696851} a^{24} - \frac{8683192}{696851} a^{23} + \frac{14575358}{696851} a^{22} - \frac{27600301}{696851} a^{21} + \frac{48067670}{696851} a^{20} - \frac{88692604}{696851} a^{19} - \frac{40935048}{696851} a^{18} - \frac{88382490}{696851} a^{17} - \frac{82180210}{696851} a^{16} - \frac{135519818}{696851} a^{15} - \frac{117223092}{696851} a^{14} - \frac{235996754}{696851} a^{13} - \frac{133969248}{696851} a^{12} - \frac{455247352}{696851} a^{11} - \frac{48790154}{696851} a^{10} - \frac{995776054}{696851} a^{9} + \frac{443152906}{696851} a^{8} - \frac{197232504}{696851} a^{7} + \frac{87762262}{696851} a^{6} - \frac{39074364}{696851} a^{5} + \frac{17366384}{696851} a^{4} - \frac{7752850}{696851} a^{3} + \frac{3411254}{696851} a^{2} - \frac{1550570}{696851} a + \frac{620228}{696851} \) (order $22$)
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

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{15}\cdot 4697581952.048968 \cdot 16}{22\sqrt{1046076147688308987260717152173116396995512371}}\approx 0.0991946028104771$ (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

\(\Q(\sqrt{-11}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{11})^+\), 6.0.3195731.1, \(\Q(\zeta_{11})\), 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 $30$ $15^{2}$ $15^{2}$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{3}$ $30$ $30$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{3}$ $15^{2}$ $15^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{15}$ $15^{2}$ $15^{2}$ $15^{2}$

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
7Data not computed
11Data not computed