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Properties

Label 36.0.42497246625...0625.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{54}\cdot 5^{18}\cdot 7^{24}$
Root discriminant $42.52$
Ramified primes $3, 5, 7$
Class number $148$ (GRH)
Class group $[2, 74]$ (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, 0, 44, 0, 0, 2129, 0, 0, -8692, 0, 0, 32635, 0, 0, -38116, 0, 0, 50596, 0, 0, 27532, 0, 0, 44003, 0, 0, 3624, 0, 0, 470, 0, 0, -16, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1)
 
gp: K = bnfinit(x^36 - 16*x^33 + 470*x^30 + 3624*x^27 + 44003*x^24 + 27532*x^21 + 50596*x^18 - 38116*x^15 + 32635*x^12 - 8692*x^9 + 2129*x^6 + 44*x^3 + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 16 x^{33} + 470 x^{30} + 3624 x^{27} + 44003 x^{24} + 27532 x^{21} + 50596 x^{18} - 38116 x^{15} + 32635 x^{12} - 8692 x^{9} + 2129 x^{6} + 44 x^{3} + 1 \)

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:  \(42497246625894555234552515412349089862939543796539306640625=3^{54}\cdot 5^{18}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.52$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(4,·)$, $\chi_{315}(86,·)$, $\chi_{315}(134,·)$, $\chi_{315}(11,·)$, $\chi_{315}(16,·)$, $\chi_{315}(274,·)$, $\chi_{315}(149,·)$, $\chi_{315}(151,·)$, $\chi_{315}(281,·)$, $\chi_{315}(284,·)$, $\chi_{315}(29,·)$, $\chi_{315}(289,·)$, $\chi_{315}(296,·)$, $\chi_{315}(169,·)$, $\chi_{315}(44,·)$, $\chi_{315}(46,·)$, $\chi_{315}(176,·)$, $\chi_{315}(179,·)$, $\chi_{315}(184,·)$, $\chi_{315}(191,·)$, $\chi_{315}(64,·)$, $\chi_{315}(71,·)$, $\chi_{315}(74,·)$, $\chi_{315}(79,·)$, $\chi_{315}(211,·)$, $\chi_{315}(214,·)$, $\chi_{315}(221,·)$, $\chi_{315}(226,·)$, $\chi_{315}(106,·)$, $\chi_{315}(109,·)$, $\chi_{315}(239,·)$, $\chi_{315}(116,·)$, $\chi_{315}(121,·)$, $\chi_{315}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5}$, $\frac{1}{26} a^{24} + \frac{3}{26} a^{21} - \frac{1}{13} a^{18} + \frac{3}{26} a^{15} + \frac{6}{13} a^{12} + \frac{1}{13} a^{9} - \frac{11}{26} a^{6} - \frac{1}{13} a^{3} + \frac{1}{26}$, $\frac{1}{26} a^{25} + \frac{3}{26} a^{22} - \frac{1}{13} a^{19} + \frac{3}{26} a^{16} + \frac{6}{13} a^{13} + \frac{1}{13} a^{10} - \frac{11}{26} a^{7} - \frac{1}{13} a^{4} + \frac{1}{26} a$, $\frac{1}{26} a^{26} + \frac{3}{26} a^{23} - \frac{1}{13} a^{20} + \frac{3}{26} a^{17} + \frac{6}{13} a^{14} + \frac{1}{13} a^{11} - \frac{11}{26} a^{8} - \frac{1}{13} a^{5} + \frac{1}{26} a^{2}$, $\frac{1}{442} a^{27} - \frac{3}{13} a^{21} - \frac{95}{442} a^{18} + \frac{2}{13} a^{15} - \frac{1}{13} a^{12} + \frac{50}{221} a^{9} - \frac{1}{13} a^{6} - \frac{4}{13} a^{3} - \frac{81}{442}$, $\frac{1}{442} a^{28} - \frac{3}{13} a^{22} - \frac{95}{442} a^{19} + \frac{2}{13} a^{16} - \frac{1}{13} a^{13} + \frac{50}{221} a^{10} - \frac{1}{13} a^{7} - \frac{4}{13} a^{4} - \frac{81}{442} a$, $\frac{1}{442} a^{29} - \frac{3}{13} a^{23} - \frac{95}{442} a^{20} + \frac{2}{13} a^{17} - \frac{1}{13} a^{14} + \frac{50}{221} a^{11} - \frac{1}{13} a^{8} - \frac{4}{13} a^{5} - \frac{81}{442} a^{2}$, $\frac{1}{7360037698} a^{30} - \frac{1683834}{3680018849} a^{27} + \frac{6441575}{432943394} a^{24} - \frac{716190659}{3680018849} a^{21} + \frac{518929651}{3680018849} a^{18} + \frac{13028009}{33303338} a^{15} - \frac{1803481330}{3680018849} a^{12} - \frac{1661750423}{3680018849} a^{9} - \frac{35361279}{432943394} a^{6} - \frac{2282011469}{7360037698} a^{3} + \frac{241393413}{7360037698}$, $\frac{1}{7360037698} a^{31} - \frac{1683834}{3680018849} a^{28} + \frac{6441575}{432943394} a^{25} - \frac{716190659}{3680018849} a^{22} + \frac{518929651}{3680018849} a^{19} + \frac{13028009}{33303338} a^{16} - \frac{1803481330}{3680018849} a^{13} - \frac{1661750423}{3680018849} a^{10} - \frac{35361279}{432943394} a^{7} - \frac{2282011469}{7360037698} a^{4} + \frac{241393413}{7360037698} a$, $\frac{1}{7360037698} a^{32} - \frac{1683834}{3680018849} a^{29} + \frac{6441575}{432943394} a^{26} - \frac{716190659}{3680018849} a^{23} + \frac{518929651}{3680018849} a^{20} + \frac{13028009}{33303338} a^{17} - \frac{1803481330}{3680018849} a^{14} - \frac{1661750423}{3680018849} a^{11} - \frac{35361279}{432943394} a^{8} - \frac{2282011469}{7360037698} a^{5} + \frac{241393413}{7360037698} a^{2}$, $\frac{1}{12287620038761035618} a^{33} - \frac{833324575}{12287620038761035618} a^{30} + \frac{1527523822460443}{12287620038761035618} a^{27} - \frac{51059455414921331}{12287620038761035618} a^{24} - \frac{687369206163464139}{12287620038761035618} a^{21} - \frac{76435553572800713}{6143810019380517809} a^{18} + \frac{1957032277557465139}{6143810019380517809} a^{15} + \frac{4429080406490146237}{12287620038761035618} a^{12} - \frac{1373115964025814875}{6143810019380517809} a^{9} - \frac{1707008940498723556}{6143810019380517809} a^{6} - \frac{697979026570989760}{6143810019380517809} a^{3} + \frac{177893123681331448}{472600770721578293}$, $\frac{1}{12287620038761035618} a^{34} - \frac{833324575}{12287620038761035618} a^{31} + \frac{1527523822460443}{12287620038761035618} a^{28} - \frac{51059455414921331}{12287620038761035618} a^{25} - \frac{687369206163464139}{12287620038761035618} a^{22} - \frac{76435553572800713}{6143810019380517809} a^{19} + \frac{1957032277557465139}{6143810019380517809} a^{16} + \frac{4429080406490146237}{12287620038761035618} a^{13} - \frac{1373115964025814875}{6143810019380517809} a^{10} - \frac{1707008940498723556}{6143810019380517809} a^{7} - \frac{697979026570989760}{6143810019380517809} a^{4} + \frac{177893123681331448}{472600770721578293} a$, $\frac{1}{12287620038761035618} a^{35} - \frac{833324575}{12287620038761035618} a^{32} + \frac{1527523822460443}{12287620038761035618} a^{29} - \frac{51059455414921331}{12287620038761035618} a^{26} - \frac{687369206163464139}{12287620038761035618} a^{23} - \frac{76435553572800713}{6143810019380517809} a^{20} + \frac{1957032277557465139}{6143810019380517809} a^{17} + \frac{4429080406490146237}{12287620038761035618} a^{14} - \frac{1373115964025814875}{6143810019380517809} a^{11} - \frac{1707008940498723556}{6143810019380517809} a^{8} - \frac{697979026570989760}{6143810019380517809} a^{5} + \frac{177893123681331448}{472600770721578293} a^{2}$

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

Class group and class number

$C_{2}\times C_{74}$, which has order $148$ (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:  \( -\frac{54750183643159558}{6143810019380517809} a^{34} + \frac{863351337584628236}{6143810019380517809} a^{31} - \frac{51059596888771568813}{12287620038761035618} a^{28} - \frac{204366554761057937494}{6143810019380517809} a^{25} - \frac{2454854100752371198598}{6143810019380517809} a^{22} - \frac{4125402857242667211415}{12287620038761035618} a^{19} - \frac{3101986311157413131518}{6143810019380517809} a^{16} + \frac{1463441623126818098554}{6143810019380517809} a^{13} - \frac{1285209272019518159592}{6143810019380517809} a^{10} + \frac{57740314069469615414}{6143810019380517809} a^{7} + \frac{1187719518152058308}{6143810019380517809} a^{4} - \frac{44600832139091345309}{12287620038761035618} a \) (order $18$)
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:  \( 13624539961495.691 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{9})\), 6.0.47258883.1, 6.0.64827.1, 6.0.47258883.2, 6.0.2460375.1, 6.6.820125.1, 6.0.5907360375.2, 6.6.1969120125.2, 6.0.8103375.1, 6.6.300125.1, 6.0.5907360375.1, 6.6.1969120125.1, 9.9.62523502209.1, 12.0.6053445140625.1, 12.0.34896906600120140625.2, 12.0.65664686390625.1, 12.0.34896906600120140625.1, 18.0.105548084868928352751387.1, 18.0.206148603259625688967552734375.1, 18.18.7635133454060210702501953125.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{6}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{6}$

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
3Data not computed
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed