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Properties

Label 36.0.123...304.1
Degree $36$
Signature $[0, 18]$
Discriminant $1.235\times 10^{59}$
Root discriminant \(43.80\)
Ramified primes $2,3,7$
Class number $196$ (GRH)
Class group [14, 14] (GRH)
Galois group $C_6^2$ (as 36T4)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1)
 
gp: K = bnfinit(y^36 - 6*y^34 + 27*y^32 - 109*y^30 + 417*y^28 - 1548*y^26 + 5644*y^24 - 13098*y^22 + 29340*y^20 - 63802*y^18 + 131850*y^16 - 246222*y^14 + 354484*y^12 - 42756*y^10 + 5157*y^8 - 622*y^6 + 75*y^4 - 9*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1)
 

\( x^{36} - 6 x^{34} + 27 x^{32} - 109 x^{30} + 417 x^{28} - 1548 x^{26} + 5644 x^{24} - 13098 x^{22} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $36$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 18]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(123549579287202724195633555037990063416945072951206088802304\) \(\medspace = 2^{36}\cdot 3^{48}\cdot 7^{30}\) 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:  \(43.80\)
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:  $2\cdot 3^{4/3}7^{5/6}\approx 43.796563540728876$
Ramified primes:   \(2\), \(3\), \(7\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $36$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(139,·)$, $\chi_{252}(13,·)$, $\chi_{252}(145,·)$, $\chi_{252}(19,·)$, $\chi_{252}(151,·)$, $\chi_{252}(25,·)$, $\chi_{252}(157,·)$, $\chi_{252}(31,·)$, $\chi_{252}(163,·)$, $\chi_{252}(37,·)$, $\chi_{252}(169,·)$, $\chi_{252}(43,·)$, $\chi_{252}(181,·)$, $\chi_{252}(55,·)$, $\chi_{252}(187,·)$, $\chi_{252}(61,·)$, $\chi_{252}(193,·)$, $\chi_{252}(67,·)$, $\chi_{252}(199,·)$, $\chi_{252}(73,·)$, $\chi_{252}(205,·)$, $\chi_{252}(79,·)$, $\chi_{252}(211,·)$, $\chi_{252}(85,·)$, $\chi_{252}(223,·)$, $\chi_{252}(97,·)$, $\chi_{252}(229,·)$, $\chi_{252}(103,·)$, $\chi_{252}(235,·)$, $\chi_{252}(109,·)$, $\chi_{252}(241,·)$, $\chi_{252}(115,·)$, $\chi_{252}(247,·)$, $\chi_{252}(121,·)$, $\chi_{252}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{131072}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4}a^{14}+\frac{1}{4}$, $\frac{1}{4}a^{15}+\frac{1}{4}a$, $\frac{1}{4}a^{16}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{17}+\frac{1}{4}a^{3}$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{19}+\frac{1}{4}a^{5}$, $\frac{1}{4}a^{20}+\frac{1}{4}a^{6}$, $\frac{1}{4}a^{21}+\frac{1}{4}a^{7}$, $\frac{1}{4}a^{22}+\frac{1}{4}a^{8}$, $\frac{1}{4}a^{23}+\frac{1}{4}a^{9}$, $\frac{1}{4}a^{24}+\frac{1}{4}a^{10}$, $\frac{1}{4}a^{25}+\frac{1}{4}a^{11}$, $\frac{1}{120907588}a^{26}-\frac{156215}{120907588}a^{24}+\frac{5010173}{60453794}a^{22}+\frac{434413}{30226897}a^{20}-\frac{2443030}{30226897}a^{18}+\frac{2746609}{120907588}a^{16}+\frac{2319495}{30226897}a^{14}-\frac{35303995}{120907588}a^{12}+\frac{55767481}{120907588}a^{10}-\frac{26643471}{60453794}a^{8}+\frac{8619925}{30226897}a^{6}-\frac{13776319}{30226897}a^{4}+\frac{43374401}{120907588}a^{2}+\frac{4964999}{30226897}$, $\frac{1}{120907588}a^{27}-\frac{156215}{120907588}a^{25}+\frac{5010173}{60453794}a^{23}+\frac{434413}{30226897}a^{21}-\frac{2443030}{30226897}a^{19}+\frac{2746609}{120907588}a^{17}+\frac{2319495}{30226897}a^{15}-\frac{35303995}{120907588}a^{13}+\frac{55767481}{120907588}a^{11}-\frac{26643471}{60453794}a^{9}+\frac{8619925}{30226897}a^{7}-\frac{13776319}{30226897}a^{5}+\frac{43374401}{120907588}a^{3}+\frac{4964999}{30226897}a$, $\frac{1}{483630352}a^{28}+\frac{17655623}{241815176}a^{14}+\frac{212609133}{483630352}$, $\frac{1}{483630352}a^{29}+\frac{17655623}{241815176}a^{15}+\frac{212609133}{483630352}a$, $\frac{1}{483630352}a^{30}+\frac{17655623}{241815176}a^{16}+\frac{212609133}{483630352}a^{2}$, $\frac{1}{483630352}a^{31}+\frac{17655623}{241815176}a^{17}+\frac{212609133}{483630352}a^{3}$, $\frac{1}{483630352}a^{32}+\frac{17655623}{241815176}a^{18}+\frac{212609133}{483630352}a^{4}$, $\frac{1}{483630352}a^{33}+\frac{17655623}{241815176}a^{19}+\frac{212609133}{483630352}a^{5}$, $\frac{1}{483630352}a^{34}+\frac{17655623}{241815176}a^{20}+\frac{212609133}{483630352}a^{6}$, $\frac{1}{483630352}a^{35}+\frac{17655623}{241815176}a^{21}+\frac{212609133}{483630352}a^{7}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{14}\times C_{14}$, which has order $196$ (assuming GRH)

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:  $17$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{35525}{483630352} a^{33} - \frac{128801913}{241815176} a^{19} - \frac{95722144749}{483630352} a^{5} \)  (order $28$) 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:   $\frac{58230615}{483630352}a^{34}-\frac{174998871}{241815176}a^{32}+\frac{1574989839}{483630352}a^{30}-\frac{6358292313}{483630352}a^{28}+\frac{1520302716}{30226897}a^{26}-\frac{22574854359}{120907588}a^{24}+\frac{82307802327}{120907588}a^{22}-\frac{382393578357}{241815176}a^{20}+\frac{427872239595}{120907588}a^{18}-\frac{1860879661257}{241815176}a^{16}+\frac{3845600190225}{241815176}a^{14}-\frac{897678402731}{30226897}a^{12}+\frac{5169524982297}{120907588}a^{10}-\frac{623520977373}{120907588}a^{8}+\frac{25202128367}{483630352}a^{6}-\frac{18141549627}{241815176}a^{4}+\frac{4374971775}{483630352}a^{2}-\frac{524996613}{483630352}$, $\frac{58332957}{483630352}a^{34}-\frac{174998871}{241815176}a^{32}+\frac{1574989839}{483630352}a^{30}-\frac{6358292313}{483630352}a^{28}+\frac{1520302716}{30226897}a^{26}-\frac{22574854359}{120907588}a^{24}+\frac{82307802327}{120907588}a^{22}-\frac{382022535393}{241815176}a^{20}+\frac{427872239595}{120907588}a^{18}-\frac{1860879661257}{241815176}a^{16}+\frac{3845600190225}{241815176}a^{14}-\frac{897678402731}{30226897}a^{12}+\frac{5169524982297}{120907588}a^{10}-\frac{623520977373}{120907588}a^{8}+\frac{300823059249}{483630352}a^{6}-\frac{18141549627}{241815176}a^{4}+\frac{4374971775}{483630352}a^{2}-\frac{41366261}{483630352}$, $\frac{7330527}{241815176}a^{35}-\frac{51171}{241815176}a^{34}-\frac{65974743}{483630352}a^{33}+\frac{266342481}{483630352}a^{31}-\frac{1018943253}{483630352}a^{29}+\frac{945637983}{120907588}a^{27}-\frac{3447791199}{120907588}a^{25}+\frac{6215370813}{60453794}a^{23}-\frac{17923138515}{120907588}a^{21}-\frac{92760741}{60453794}a^{20}+\frac{77950380609}{241815176}a^{19}-\frac{161088330825}{241815176}a^{17}+\frac{300822836499}{241815176}a^{15}-\frac{216546211089}{120907588}a^{13}+\frac{26118667701}{120907588}a^{11}+\frac{820897628101}{60453794}a^{9}+\frac{759931299}{241815176}a^{7}-\frac{137810465441}{241815176}a^{6}-\frac{183263175}{483630352}a^{5}+\frac{21991581}{483630352}a^{3}-\frac{2443509}{483630352}a$, $\frac{387}{483630352}a^{29}+\frac{1447379}{241815176}a^{15}+\frac{1392558099}{483630352}a-1$, $\frac{7330527}{241815176}a^{35}-\frac{65974743}{483630352}a^{33}+\frac{266342481}{483630352}a^{31}-\frac{1018943253}{483630352}a^{29}+\frac{945637983}{120907588}a^{27}-\frac{3447791199}{120907588}a^{25}+\frac{6215370813}{60453794}a^{23}-\frac{17923138515}{120907588}a^{21}+\frac{77950380609}{241815176}a^{19}-\frac{161088330825}{241815176}a^{17}+\frac{300822836499}{241815176}a^{15}-\frac{216546211089}{120907588}a^{13}+\frac{26118667701}{120907588}a^{11}+\frac{820897628101}{60453794}a^{9}+\frac{759931299}{241815176}a^{7}-\frac{183263175}{483630352}a^{5}+\frac{21991581}{483630352}a^{3}-\frac{2443509}{483630352}a+1$, $\frac{1539}{60453794}a^{31}+\frac{22320473}{120907588}a^{17}+\frac{8310987999}{120907588}a^{3}$, $\frac{64}{30226897}a^{30}+\frac{1836773}{120907588}a^{16}+\frac{654288285}{120907588}a^{2}$, $\frac{315113391}{483630352}a^{35}-\frac{473500941}{120907588}a^{33}+\frac{4261502313}{241815176}a^{31}-\frac{17203867523}{241815176}a^{29}+\frac{32908315337}{120907588}a^{27}-\frac{61081621389}{60453794}a^{25}+\frac{222703275917}{60453794}a^{23}-\frac{2069313282385}{241815176}a^{21}+\frac{1157709800745}{60453794}a^{19}-\frac{1258768373355}{30226897}a^{17}+\frac{10405183178475}{120907588}a^{15}-\frac{19431058505211}{120907588}a^{13}+\frac{13987375630787}{60453794}a^{11}-\frac{1687083852783}{60453794}a^{9}+\frac{136380698607}{483630352}a^{7}-\frac{49086264217}{120907588}a^{5}-\frac{4784452473}{241815176}a^{3}-\frac{1420502823}{241815176}a$, $\frac{91164699}{241815176}a^{34}-\frac{1073753091}{483630352}a^{32}+\frac{2400670407}{241815176}a^{30}-\frac{4831729047}{120907588}a^{28}+\frac{9227893421}{60453794}a^{26}-\frac{68449494645}{120907588}a^{24}+\frac{62356654116}{30226897}a^{22}-\frac{568452415909}{120907588}a^{20}+\frac{2541968441469}{241815176}a^{18}-\frac{2759646603429}{120907588}a^{16}+\frac{1421723610316}{30226897}a^{14}-\frac{5277797919207}{60453794}a^{12}+\frac{14911172830719}{120907588}a^{10}-\frac{38390467690}{30226897}a^{8}+\frac{37043256027}{241815176}a^{6}-\frac{104656285251}{483630352}a^{4}+\frac{536858783}{241815176}a^{2}-\frac{151295821}{120907588}$, $\frac{78843145}{120907588}a^{35}-\frac{1893968239}{483630352}a^{33}+\frac{4261508469}{241815176}a^{31}-\frac{17203867523}{241815176}a^{29}+\frac{32908315337}{120907588}a^{27}-\frac{61081621389}{60453794}a^{25}+\frac{222703275917}{60453794}a^{23}-\frac{258546699544}{30226897}a^{21}+\frac{4630968004893}{241815176}a^{19}-\frac{5035051172947}{120907588}a^{17}+\frac{10405183178475}{120907588}a^{15}-\frac{19431058505211}{120907588}a^{13}+\frac{13987375630787}{60453794}a^{11}-\frac{1687083852783}{60453794}a^{9}+\frac{208569348627}{120907588}a^{7}-\frac{100622912119}{483630352}a^{5}+\frac{11837523525}{241815176}a^{3}-\frac{1420502823}{241815176}a$, $\frac{3690849}{120907588}a^{35}-\frac{32969609}{241815176}a^{33}+\frac{266342481}{483630352}a^{31}-\frac{1018943253}{483630352}a^{29}+\frac{945637983}{120907588}a^{27}-\frac{3447791199}{120907588}a^{25}+\frac{6215370813}{60453794}a^{23}-\frac{17737617033}{120907588}a^{21}+\frac{39039591261}{120907588}a^{19}-\frac{161088330825}{241815176}a^{17}+\frac{300822836499}{241815176}a^{15}-\frac{216546211089}{120907588}a^{13}+\frac{26118667701}{120907588}a^{11}+\frac{820897628101}{60453794}a^{9}+\frac{34642599185}{60453794}a^{7}+\frac{47769440787}{241815176}a^{5}+\frac{21991581}{483630352}a^{3}-\frac{2443509}{483630352}a$, $\frac{91164699}{241815176}a^{35}+\frac{156847}{483630352}a^{34}-\frac{536858783}{241815176}a^{33}+\frac{2400670407}{241815176}a^{31}-\frac{19326915801}{483630352}a^{29}+\frac{9227893421}{60453794}a^{27}-\frac{68449494645}{120907588}a^{25}+\frac{62356654116}{30226897}a^{23}-\frac{568452415909}{120907588}a^{21}+\frac{568643069}{241815176}a^{20}+\frac{1271048621691}{120907588}a^{19}-\frac{2759646603429}{120907588}a^{17}+\frac{11373790329907}{241815176}a^{15}-\frac{5277797919207}{60453794}a^{13}+\frac{14911172830719}{120907588}a^{11}-\frac{38390467690}{30226897}a^{9}+\frac{37043256027}{241815176}a^{7}+\frac{422275765019}{483630352}a^{6}-\frac{4467070251}{241815176}a^{5}+\frac{536858783}{241815176}a^{3}+\frac{1271005167}{483630352}a$, $\frac{58332957}{483630352}a^{34}-\frac{175005027}{241815176}a^{32}+\frac{1574989839}{483630352}a^{30}+\frac{64}{30226897}a^{29}-\frac{6358292313}{483630352}a^{28}+\frac{1520302716}{30226897}a^{26}-\frac{22574854359}{120907588}a^{24}+\frac{82307802327}{120907588}a^{22}-\frac{382022535393}{241815176}a^{20}+\frac{213924959561}{60453794}a^{18}-\frac{1860879661257}{241815176}a^{16}+\frac{1836773}{120907588}a^{15}+\frac{3845600190225}{241815176}a^{14}-\frac{897678402731}{30226897}a^{12}+\frac{5169524982297}{120907588}a^{10}-\frac{623520977373}{120907588}a^{8}+\frac{300823059249}{483630352}a^{6}-\frac{34763525625}{241815176}a^{4}+\frac{4374971775}{483630352}a^{2}+\frac{654288285}{120907588}a-\frac{524996613}{483630352}$, $\frac{42064443}{120907588}a^{35}-\frac{21991581}{483630352}a^{34}-\frac{251944587}{120907588}a^{33}+\frac{129505977}{483630352}a^{32}+\frac{2267501283}{241815176}a^{31}-\frac{579111633}{483630352}a^{30}-\frac{9153986661}{241815176}a^{29}+\frac{1165553793}{241815176}a^{28}+\frac{17510148859}{120907588}a^{27}-\frac{2226036699}{120907588}a^{26}-\frac{32500851723}{60453794}a^{25}+\frac{2064001392}{30226897}a^{24}+\frac{118497937419}{60453794}a^{23}-\frac{7521120702}{30226897}a^{22}-\frac{137365197362}{30226897}a^{21}+\frac{137127281571}{241815176}a^{20}+\frac{616004515215}{60453794}a^{19}-\frac{306613952829}{241815176}a^{18}-\frac{2679094756629}{120907588}a^{17}+\frac{665707148451}{241815176}a^{16}+\frac{5536482299325}{120907588}a^{15}-\frac{171480574602}{30226897}a^{14}-\frac{10339049627325}{120907588}a^{13}+\frac{1273158598833}{120907588}a^{12}+\frac{7442527081509}{60453794}a^{11}-\frac{449626815602}{30226897}a^{10}-\frac{897678563481}{60453794}a^{9}+\frac{4630449555}{30226897}a^{8}+\frac{414951082689}{120907588}a^{7}-\frac{8935912413}{483630352}a^{6}-\frac{26118255519}{120907588}a^{5}+\frac{1077587469}{483630352}a^{4}+\frac{6298614675}{241815176}a^{3}-\frac{129505977}{483630352}a^{2}-\frac{755833761}{241815176}a+\frac{7330527}{241815176}$, $\frac{83981529}{241815176}a^{35}+\frac{189}{51692}a^{34}-\frac{251944587}{120907588}a^{33}-\frac{1701}{103384}a^{32}+\frac{4535006799}{483630352}a^{31}+\frac{6867}{103384}a^{30}-\frac{9153986661}{241815176}a^{29}-\frac{26271}{103384}a^{28}+\frac{17510148859}{120907588}a^{27}+\frac{24381}{25846}a^{26}-\frac{32500851723}{60453794}a^{25}-\frac{88893}{25846}a^{24}+\frac{118497937419}{60453794}a^{23}+\frac{320497}{25846}a^{22}-\frac{549995033421}{120907588}a^{21}-\frac{462105}{25846}a^{20}+\frac{616004515215}{60453794}a^{19}+\frac{2009763}{51692}a^{18}-\frac{5358174150483}{241815176}a^{17}-\frac{4153275}{51692}a^{16}+\frac{5536482299325}{120907588}a^{15}+\frac{7755993}{51692}a^{14}-\frac{10339049627325}{120907588}a^{13}-\frac{5583123}{25846}a^{12}+\frac{7442527081509}{60453794}a^{11}+\frac{673407}{25846}a^{10}-\frac{897678563481}{60453794}a^{9}+\frac{42331071}{25846}a^{8}+\frac{433092745053}{241815176}a^{7}+\frac{19593}{51692}a^{6}-\frac{26118255519}{120907588}a^{5}-\frac{4725}{103384}a^{4}+\frac{24142732715}{483630352}a^{3}+\frac{567}{103384}a^{2}-\frac{755833761}{241815176}a-\frac{63}{103384}$, $\frac{51171}{241815176}a^{35}-\frac{1298511}{120907588}a^{34}+\frac{22912659}{483630352}a^{32}-\frac{92499253}{483630352}a^{30}+\frac{353873289}{483630352}a^{28}-\frac{328414779}{120907588}a^{26}+\frac{1197398587}{120907588}a^{24}-\frac{4317127329}{120907588}a^{22}+\frac{92760741}{60453794}a^{21}+\frac{6039084213}{120907588}a^{20}-\frac{27071730917}{241815176}a^{18}+\frac{55945075725}{241815176}a^{16}-\frac{104474087487}{241815176}a^{14}+\frac{75205287157}{120907588}a^{12}-\frac{9070867113}{120907588}a^{10}-\frac{570189514833}{120907588}a^{8}+\frac{137810465441}{241815176}a^{7}-\frac{17259298166}{30226897}a^{6}+\frac{63646275}{483630352}a^{4}-\frac{7637553}{483630352}a^{2}+\frac{848617}{483630352}$, $\frac{43786557}{60453794}a^{35}-\frac{109630101}{483630352}a^{34}-\frac{1040747957}{241815176}a^{33}+\frac{328890303}{241815176}a^{32}+\frac{2334085845}{120907588}a^{31}-\frac{2960012727}{483630352}a^{30}-\frac{18817444755}{241815176}a^{29}+\frac{11949681009}{483630352}a^{28}+\frac{35965935701}{120907588}a^{27}-\frac{11428937995}{120907588}a^{26}-\frac{133451198091}{120907588}a^{25}+\frac{42426849087}{120907588}a^{24}+\frac{243211245651}{60453794}a^{23}-\frac{154688072511}{120907588}a^{22}-\frac{559223724665}{60453794}a^{21}+\frac{717967531449}{241815176}a^{20}+\frac{2503057652121}{120907588}a^{19}-\frac{804136790835}{120907588}a^{18}-\frac{2719370680029}{60453794}a^{17}+\frac{3497309852001}{241815176}a^{16}+\frac{11223376740589}{120907588}a^{15}-\frac{7227364408425}{241815176}a^{14}-\frac{20894645465739}{120907588}a^{13}+\frac{6748336016401}{120907588}a^{12}+\frac{29796226993737}{120907588}a^{11}-\frac{9715529180721}{120907588}a^{10}-\frac{974459498861}{60453794}a^{9}+\frac{1171836149589}{120907588}a^{8}+\frac{58767000135}{30226897}a^{7}-\frac{565362430857}{483630352}a^{6}-\frac{56703581289}{241815176}a^{5}+\frac{34094961411}{241815176}a^{4}+\frac{3417736729}{120907588}a^{3}-\frac{8222257575}{483630352}a^{2}-\frac{816610227}{241815176}a+\frac{986670909}{483630352}$ Copy content Toggle raw display (assuming GRH)
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:  \( 57365817603378.39 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 57365817603378.39 \cdot 196}{28\cdot\sqrt{123549579287202724195633555037990063416945072951206088802304}}\cr\approx \mathstrut & 0.266114306242192 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1)
 
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^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1, 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^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1);
 
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^36 - 6*x^34 + 27*x^32 - 109*x^30 + 417*x^28 - 1548*x^26 + 5644*x^24 - 13098*x^22 + 29340*x^20 - 63802*x^18 + 131850*x^16 - 246222*x^14 + 354484*x^12 - 42756*x^10 + 5157*x^8 - 622*x^6 + 75*x^4 - 9*x^2 + 1);
 
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

$C_6^2$ (as 36T4):

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)
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, \(\Q(i, \sqrt{7})\), 6.0.419904.1, 6.0.1008189504.2, 6.0.153664.1, 6.0.1008189504.1, 6.6.144027072.1, 6.0.2250423.1, 6.6.7057326528.1, 6.0.110270727.1, \(\Q(\zeta_{28})^+\), \(\Q(\zeta_{7})\), 6.6.7057326528.2, 6.0.110270727.2, 9.9.62523502209.1, 12.0.20743797468893184.1, 12.0.49805857722812534784.1, \(\Q(\zeta_{28})\), 12.0.49805857722812534784.2, 18.0.1024770265180753855691096064.1, 18.18.351496200956998572502045949952.1, 18.0.1340851596668237962730583.1

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]
 

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 ${\href{/padicField/5.6.0.1}{6} }^{6}$ R ${\href{/padicField/11.6.0.1}{6} }^{6}$ ${\href{/padicField/13.6.0.1}{6} }^{6}$ ${\href{/padicField/17.6.0.1}{6} }^{6}$ ${\href{/padicField/19.6.0.1}{6} }^{6}$ ${\href{/padicField/23.6.0.1}{6} }^{6}$ ${\href{/padicField/29.3.0.1}{3} }^{12}$ ${\href{/padicField/31.6.0.1}{6} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{12}$ ${\href{/padicField/41.6.0.1}{6} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{6}$ ${\href{/padicField/47.6.0.1}{6} }^{6}$ ${\href{/padicField/53.3.0.1}{3} }^{12}$ ${\href{/padicField/59.6.0.1}{6} }^{6}$

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

# 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 2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$$2$$3$$6$$C_6$$[2]^{3}$
\(3\) Copy content Toggle raw display Deg $18$$3$$6$$24$
Deg $18$$3$$6$$24$
\(7\) Copy content Toggle raw display Deg $36$$6$$6$$30$