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Properties

Label 36.0.84073959211...0304.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 7^{30}\cdot 13^{24}$
Root discriminant $55.96$
Ramified primes $2, 7, 13$
Class number $1404$ (GRH)
Class group $[3, 6, 78]$ (GRH)
Galois group $C_6^2$ (as 36T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -14, 0, 187, 0, -2493, 0, 33233, 0, -443012, 0, 5905564, 0, -3920642, 0, 2181924, 0, -1166722, 0, 617434, 0, -325502, 0, 166844, 0, -23732, 0, 3373, 0, -478, 0, 67, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*x^2 + 1)
 
gp: K = bnfinit(x^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 9 x^{34} + 67 x^{32} - 478 x^{30} + 3373 x^{28} - 23732 x^{26} + 166844 x^{24} - 325502 x^{22} + 617434 x^{20} - 1166722 x^{18} + 2181924 x^{16} - 3920642 x^{14} + 5905564 x^{12} - 443012 x^{10} + 33233 x^{8} - 2493 x^{6} + 187 x^{4} - 14 x^{2} + 1 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(840739592110096304569466677876458557457237523263143378686050304=2^{36}\cdot 7^{30}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.96$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(364=2^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(3,·)$, $\chi_{364}(261,·)$, $\chi_{364}(263,·)$, $\chi_{364}(9,·)$, $\chi_{364}(139,·)$, $\chi_{364}(269,·)$, $\chi_{364}(131,·)$, $\chi_{364}(27,·)$, $\chi_{364}(29,·)$, $\chi_{364}(159,·)$, $\chi_{364}(289,·)$, $\chi_{364}(165,·)$, $\chi_{364}(295,·)$, $\chi_{364}(157,·)$, $\chi_{364}(243,·)$, $\chi_{364}(53,·)$, $\chi_{364}(183,·)$, $\chi_{364}(185,·)$, $\chi_{364}(87,·)$, $\chi_{364}(61,·)$, $\chi_{364}(191,·)$, $\chi_{364}(321,·)$, $\chi_{364}(55,·)$, $\chi_{364}(79,·)$, $\chi_{364}(209,·)$, $\chi_{364}(211,·)$, $\chi_{364}(341,·)$, $\chi_{364}(313,·)$, $\chi_{364}(347,·)$, $\chi_{364}(107,·)$, $\chi_{364}(81,·)$, $\chi_{364}(235,·)$, $\chi_{364}(237,·)$, $\chi_{364}(113,·)$, $\chi_{364}(339,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4} a^{14} + \frac{1}{4}$, $\frac{1}{4} a^{15} + \frac{1}{4} a$, $\frac{1}{4} a^{16} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{19} + \frac{1}{4} a^{5}$, $\frac{1}{4} a^{20} + \frac{1}{4} a^{6}$, $\frac{1}{4} a^{21} + \frac{1}{4} a^{7}$, $\frac{1}{4} a^{22} + \frac{1}{4} a^{8}$, $\frac{1}{4} a^{23} + \frac{1}{4} a^{9}$, $\frac{1}{20} a^{24} + \frac{1}{20} a^{20} + \frac{1}{20} a^{16} - \frac{1}{5} a^{12} + \frac{1}{4} a^{10} - \frac{1}{5} a^{8} + \frac{1}{4} a^{6} - \frac{1}{5} a^{4} + \frac{1}{4} a^{2} - \frac{1}{5}$, $\frac{1}{20} a^{25} + \frac{1}{20} a^{21} + \frac{1}{20} a^{17} - \frac{1}{5} a^{13} + \frac{1}{4} a^{11} - \frac{1}{5} a^{9} + \frac{1}{4} a^{7} - \frac{1}{5} a^{5} + \frac{1}{4} a^{3} - \frac{1}{5} a$, $\frac{1}{304623237260} a^{26} - \frac{4605546001}{304623237260} a^{24} - \frac{37413186779}{304623237260} a^{22} + \frac{20417278449}{304623237260} a^{20} - \frac{36144391709}{304623237260} a^{18} + \frac{715004559}{304623237260} a^{16} + \frac{5558577139}{76155809315} a^{14} - \frac{86986800951}{304623237260} a^{12} + \frac{62203110071}{304623237260} a^{10} - \frac{80658946771}{304623237260} a^{8} - \frac{134082028419}{304623237260} a^{6} + \frac{65241930359}{304623237260} a^{4} + \frac{75981213051}{304623237260} a^{2} - \frac{23082836599}{76155809315}$, $\frac{1}{304623237260} a^{27} - \frac{4605546001}{304623237260} a^{25} - \frac{37413186779}{304623237260} a^{23} + \frac{20417278449}{304623237260} a^{21} - \frac{36144391709}{304623237260} a^{19} + \frac{715004559}{304623237260} a^{17} + \frac{5558577139}{76155809315} a^{15} - \frac{86986800951}{304623237260} a^{13} + \frac{62203110071}{304623237260} a^{11} - \frac{80658946771}{304623237260} a^{9} - \frac{134082028419}{304623237260} a^{7} + \frac{65241930359}{304623237260} a^{5} + \frac{75981213051}{304623237260} a^{3} - \frac{23082836599}{76155809315} a$, $\frac{1}{1218492949040} a^{28} + \frac{8698764817}{121849294904} a^{14} - \frac{6626441751}{1218492949040}$, $\frac{1}{1218492949040} a^{29} + \frac{8698764817}{121849294904} a^{15} - \frac{6626441751}{1218492949040} a$, $\frac{1}{1218492949040} a^{30} + \frac{8698764817}{121849294904} a^{16} - \frac{6626441751}{1218492949040} a^{2}$, $\frac{1}{1218492949040} a^{31} + \frac{8698764817}{121849294904} a^{17} - \frac{6626441751}{1218492949040} a^{3}$, $\frac{1}{1218492949040} a^{32} + \frac{8698764817}{121849294904} a^{18} - \frac{6626441751}{1218492949040} a^{4}$, $\frac{1}{1218492949040} a^{33} + \frac{8698764817}{121849294904} a^{19} - \frac{6626441751}{1218492949040} a^{5}$, $\frac{1}{1218492949040} a^{34} + \frac{8698764817}{121849294904} a^{20} - \frac{6626441751}{1218492949040} a^{6}$, $\frac{1}{1218492949040} a^{35} + \frac{8698764817}{121849294904} a^{21} - \frac{6626441751}{1218492949040} a^{7}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{78}$, which has order $1404$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{11911}{304623237260} a^{29} + \frac{2018324459}{60924647452} a^{15} + \frac{278051973476}{76155809315} a \) (order $28$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 695961018492374.6 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6^2$ (as 36T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_6^2$
Character table for $C_6^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), 3.3.169.1, 3.3.8281.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), \(\Q(i, \sqrt{7})\), 6.0.1827904.1, 6.0.4388797504.2, 6.0.4388797504.1, 6.0.153664.1, 6.6.626971072.1, 6.0.9796423.1, 6.6.30721582528.1, 6.0.480024727.1, 6.6.30721582528.2, 6.0.480024727.2, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), 9.9.567869252041.1, 12.0.393092725124829184.1, 12.0.943815633024714870784.1, 12.0.943815633024714870784.2, \(\Q(\zeta_{28})\), 18.0.84535014172552012147112280064.1, 18.18.28995509861185340166459512061952.1, 18.0.110609092182866440454328583.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$