Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 304*x^30 + 84160*x^24 - 2508800*x^18 + 68005888*x^12 - 4227072*x^6 + 262144)
gp: K = bnfinit(y^36 - 304*y^30 + 84160*y^24 - 2508800*y^18 + 68005888*y^12 - 4227072*y^6 + 262144, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 304*x^30 + 84160*x^24 - 2508800*x^18 + 68005888*x^12 - 4227072*x^6 + 262144);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 304*x^30 + 84160*x^24 - 2508800*x^18 + 68005888*x^12 - 4227072*x^6 + 262144)
\( x^{36} - 304x^{30} + 84160x^{24} - 2508800x^{18} + 68005888x^{12} - 4227072x^{6} + 262144 \)
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: |
\(200687573080369568029416132506181048520658333428355416190877696\)
\(\medspace = 2^{54}\cdot 3^{54}\cdot 7^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(53.78\) | 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: | $2^{3/2}3^{3/2}7^{2/3}\approx 53.7805908144551$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| 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: | \(504=2^{3}\cdot 3^{2}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(107,·)$, $\chi_{504}(193,·)$, $\chi_{504}(137,·)$, $\chi_{504}(11,·)$, $\chi_{504}(401,·)$, $\chi_{504}(275,·)$, $\chi_{504}(25,·)$, $\chi_{504}(281,·)$, $\chi_{504}(155,·)$, $\chi_{504}(289,·)$, $\chi_{504}(163,·)$, $\chi_{504}(65,·)$, $\chi_{504}(169,·)$, $\chi_{504}(43,·)$, $\chi_{504}(305,·)$, $\chi_{504}(179,·)$, $\chi_{504}(443,·)$, $\chi_{504}(449,·)$, $\chi_{504}(67,·)$, $\chi_{504}(457,·)$, $\chi_{504}(331,·)$, $\chi_{504}(403,·)$, $\chi_{504}(337,·)$, $\chi_{504}(211,·)$, $\chi_{504}(473,·)$, $\chi_{504}(347,·)$, $\chi_{504}(361,·)$, $\chi_{504}(323,·)$, $\chi_{504}(491,·)$, $\chi_{504}(113,·)$, $\chi_{504}(499,·)$, $\chi_{504}(235,·)$, $\chi_{504}(233,·)$, $\chi_{504}(121,·)$, $\chi_{504}(379,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{2}a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{4}a^{4}$, $\frac{1}{4}a^{5}$, $\frac{1}{8}a^{6}$, $\frac{1}{8}a^{7}$, $\frac{1}{16}a^{8}$, $\frac{1}{16}a^{9}$, $\frac{1}{32}a^{10}$, $\frac{1}{32}a^{11}$, $\frac{1}{64}a^{12}$, $\frac{1}{64}a^{13}$, $\frac{1}{128}a^{14}$, $\frac{1}{128}a^{15}$, $\frac{1}{256}a^{16}$, $\frac{1}{256}a^{17}$, $\frac{1}{512}a^{18}$, $\frac{1}{512}a^{19}$, $\frac{1}{1024}a^{20}$, $\frac{1}{1024}a^{21}$, $\frac{1}{2048}a^{22}$, $\frac{1}{2048}a^{23}$, $\frac{1}{2289664}a^{24}-\frac{3}{5504}a^{18}+\frac{209}{35776}a^{12}-\frac{5}{344}a^{6}+\frac{274}{559}$, $\frac{1}{2289664}a^{25}-\frac{3}{5504}a^{19}+\frac{209}{35776}a^{13}-\frac{5}{344}a^{7}+\frac{274}{559}a$, $\frac{1}{4579328}a^{26}-\frac{3}{11008}a^{20}+\frac{209}{71552}a^{14}-\frac{5}{688}a^{8}+\frac{137}{559}a^{2}$, $\frac{1}{4579328}a^{27}-\frac{3}{11008}a^{21}+\frac{209}{71552}a^{15}-\frac{5}{688}a^{9}+\frac{137}{559}a^{3}$, $\frac{1}{9158656}a^{28}-\frac{3}{22016}a^{22}+\frac{209}{143104}a^{16}-\frac{5}{1376}a^{10}+\frac{137}{1118}a^{4}$, $\frac{1}{9158656}a^{29}-\frac{3}{22016}a^{23}+\frac{209}{143104}a^{17}-\frac{5}{1376}a^{11}+\frac{137}{1118}a^{5}$, $\frac{1}{399831073980416}a^{30}-\frac{7467523}{49978884247552}a^{24}-\frac{114384399}{6247360530944}a^{18}+\frac{2751769255}{390460033184}a^{12}+\frac{1127590751}{97615008296}a^{6}+\frac{5582733509}{12201876037}$, $\frac{1}{399831073980416}a^{31}-\frac{7467523}{49978884247552}a^{25}-\frac{114384399}{6247360530944}a^{19}+\frac{2751769255}{390460033184}a^{13}+\frac{1127590751}{97615008296}a^{7}+\frac{5582733509}{12201876037}a$, $\frac{1}{799662147960832}a^{32}-\frac{7467523}{99957768495104}a^{26}-\frac{114384399}{12494721061888}a^{20}+\frac{2751769255}{780920066368}a^{14}+\frac{1127590751}{195230016592}a^{8}+\frac{5582733509}{24403752074}a^{2}$, $\frac{1}{799662147960832}a^{33}-\frac{7467523}{99957768495104}a^{27}-\frac{114384399}{12494721061888}a^{21}+\frac{2751769255}{780920066368}a^{15}+\frac{1127590751}{195230016592}a^{9}+\frac{5582733509}{24403752074}a^{3}$, $\frac{1}{15\!\cdots\!64}a^{34}-\frac{7467523}{199915536990208}a^{28}-\frac{114384399}{24989442123776}a^{22}+\frac{2751769255}{1561840132736}a^{16}+\frac{1127590751}{390460033184}a^{10}+\frac{5582733509}{48807504148}a^{4}$, $\frac{1}{15\!\cdots\!64}a^{35}-\frac{7467523}{199915536990208}a^{29}-\frac{114384399}{24989442123776}a^{23}+\frac{2751769255}{1561840132736}a^{17}+\frac{1127590751}{390460033184}a^{11}+\frac{5582733509}{48807504148}a^{5}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
$C_{18}\times C_{18}$, which has order $324$ (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: |
\( -\frac{1208369401}{799662147960832} a^{34} + \frac{45917889931}{99957768495104} a^{28} - \frac{3178001329435}{24989442123776} a^{22} + \frac{11841670194505}{3123680265472} a^{16} - \frac{40124985606547}{390460033184} a^{10} + \frac{7250193147}{1135058236} a^{4} \)
(order $18$)
| 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: |
$\frac{579}{858914816}a^{34}-\frac{472947}{2308333568}a^{28}+\frac{32732221}{577083392}a^{22}-\frac{121741505}{72135424}a^{16}+\frac{206190241}{4508464}a^{10}-\frac{3667}{563558}a^{4}$, $\frac{1437141}{799662147960832}a^{34}-\frac{99465285}{199915536990208}a^{28}+\frac{1704640149}{12494721061888}a^{22}-\frac{1255834317}{3123680265472}a^{16}+\frac{226917}{9080465888}a^{10}+\frac{172616840243}{48807504148}a^{4}$, $\frac{3520629}{2324599267328}a^{34}+\frac{119669137}{199915536990208}a^{32}-\frac{92024515237}{199915536990208}a^{28}-\frac{2273649231}{12494721061888}a^{26}+\frac{199029372171}{1561840132736}a^{22}+\frac{157360459935}{3123680265472}a^{20}-\frac{740253321255}{195230016592}a^{16}-\frac{2344901835491}{1561840132736}a^{14}+\frac{5015625515159}{48807504148}a^{10}+\frac{1986810430647}{48807504148}a^{8}-\frac{178378536}{12201876037}a^{4}-\frac{717994494}{283764559}a^{2}$, $\frac{3520629}{2324599267328}a^{34}-\frac{92024515237}{199915536990208}a^{28}+\frac{199029372171}{1561840132736}a^{22}-\frac{740253321255}{195230016592}a^{16}+\frac{5015625515159}{48807504148}a^{10}-\frac{178378536}{12201876037}a^{4}-1$, $\frac{3520629}{2324599267328}a^{34}-\frac{66417139}{199915536990208}a^{32}-\frac{608043}{4649198534656}a^{30}-\frac{92024515237}{199915536990208}a^{28}+\frac{5047112139}{49978884247552}a^{26}+\frac{1986490003}{49978884247552}a^{24}+\frac{199029372171}{1561840132736}a^{22}-\frac{349313287515}{12494721061888}a^{20}-\frac{34374089557}{3123680265472}a^{18}-\frac{740253321255}{195230016592}a^{16}+\frac{1300379634991}{1561840132736}a^{14}+\frac{127848134585}{390460033184}a^{12}+\frac{5015625515159}{48807504148}a^{10}-\frac{4410379097043}{195230016592}a^{8}-\frac{873515219295}{97615008296}a^{6}-\frac{178378536}{12201876037}a^{4}+\frac{796912443}{567529118}a^{2}+\frac{15403756}{12201876037}$, $\frac{1208369401}{799662147960832}a^{34}+\frac{82935}{99957768495104}a^{32}-\frac{169635}{715261313024}a^{30}-\frac{45917889931}{99957768495104}a^{28}-\frac{5739975}{24989442123776}a^{26}+\frac{3222343}{44703832064}a^{24}+\frac{3178001329435}{24989442123776}a^{22}+\frac{786491471}{12494721061888}a^{20}-\frac{223020055}{11175958016}a^{18}-\frac{11841670194505}{3123680265472}a^{16}-\frac{72472095}{390460033184}a^{14}+\frac{829482275}{1396994752}a^{12}+\frac{40124985606547}{390460033184}a^{10}+\frac{13095}{1135058236}a^{8}-\frac{2815818991}{174624344}a^{6}-\frac{7250193147}{1135058236}a^{4}+\frac{27417349451}{24403752074}a^{2}+\frac{49970}{21828043}$, $\frac{1208369401}{799662147960832}a^{34}-\frac{119669137}{199915536990208}a^{32}-\frac{989169}{9298397069312}a^{30}-\frac{45917889931}{99957768495104}a^{28}+\frac{2273649231}{12494721061888}a^{26}+\frac{1616089471}{49978884247552}a^{24}+\frac{3178001329435}{24989442123776}a^{22}-\frac{157360459935}{3123680265472}a^{20}-\frac{55920031631}{6247360530944}a^{18}-\frac{11841670194505}{3123680265472}a^{16}+\frac{2344901835491}{1561840132736}a^{14}+\frac{207984322555}{780920066368}a^{12}+\frac{40124985606547}{390460033184}a^{10}-\frac{1986810430647}{48807504148}a^{8}-\frac{350263798337}{48807504148}a^{6}-\frac{7250193147}{1135058236}a^{4}+\frac{717994494}{283764559}a^{2}+\frac{12529474}{12201876037}$, $\frac{1208369401}{799662147960832}a^{34}-\frac{479047}{399831073980416}a^{32}-\frac{82935}{49978884247552}a^{30}-\frac{45917889931}{99957768495104}a^{28}+\frac{33155095}{99957768495104}a^{26}+\frac{5739975}{12494721061888}a^{24}+\frac{3178001329435}{24989442123776}a^{22}-\frac{568213383}{6247360530944}a^{20}-\frac{786491471}{6247360530944}a^{18}-\frac{11841670194505}{3123680265472}a^{16}+\frac{418611439}{1561840132736}a^{14}+\frac{72472095}{195230016592}a^{12}+\frac{40124985606547}{390460033184}a^{10}-\frac{75639}{4540232944}a^{8}-\frac{13095}{567529118}a^{6}-\frac{7250193147}{1135058236}a^{4}-\frac{30803119380}{12201876037}a^{2}-\frac{15215473414}{12201876037}$, $\frac{3520629}{2324599267328}a^{34}-\frac{82733471}{61512472920064}a^{33}-\frac{479047}{399831073980416}a^{32}-\frac{92024515237}{199915536990208}a^{28}+\frac{49123531}{120141548672}a^{27}+\frac{33155095}{99957768495104}a^{26}+\frac{199029372171}{1561840132736}a^{22}-\frac{3399865435}{30035387168}a^{21}-\frac{568213383}{6247360530944}a^{20}-\frac{740253321255}{195230016592}a^{16}+\frac{405434252289}{120141548672}a^{15}+\frac{418611439}{1561840132736}a^{14}+\frac{5015625515159}{48807504148}a^{10}-\frac{85852419494}{938605849}a^{9}-\frac{75639}{4540232944}a^{8}-\frac{178378536}{12201876037}a^{4}+\frac{124101552}{21828043}a^{3}-\frac{30803119380}{12201876037}a^{2}$, $\frac{5430336179}{15\!\cdots\!64}a^{35}-\frac{3520629}{2324599267328}a^{34}+\frac{169635}{715261313024}a^{30}-\frac{51588285917}{49978884247552}a^{29}+\frac{92024515237}{199915536990208}a^{28}-\frac{3222343}{44703832064}a^{24}+\frac{3570452420045}{12494721061888}a^{23}-\frac{199029372171}{1561840132736}a^{22}+\frac{223020055}{11175958016}a^{18}-\frac{6652270958319}{780920066368}a^{17}+\frac{740253321255}{195230016592}a^{16}-\frac{829482275}{1396994752}a^{12}+\frac{45080016372629}{195230016592}a^{11}-\frac{5015625515159}{48807504148}a^{10}+\frac{2815818991}{174624344}a^{6}-\frac{8145518829}{567529118}a^{5}+\frac{178378536}{12201876037}a^{4}-\frac{49970}{21828043}$, $\frac{126578271}{37193588277248}a^{35}+\frac{3520629}{2324599267328}a^{34}+\frac{169635}{715261313024}a^{30}-\frac{206786729413}{199915536990208}a^{29}-\frac{92024515237}{199915536990208}a^{28}-\frac{3222343}{44703832064}a^{24}+\frac{7155765008929}{24989442123776}a^{23}+\frac{199029372171}{1561840132736}a^{22}+\frac{223020055}{11175958016}a^{18}-\frac{26614558224245}{3123680265472}a^{17}-\frac{740253321255}{195230016592}a^{16}-\frac{829482275}{1396994752}a^{12}+\frac{90160075279525}{390460033184}a^{11}+\frac{5015625515159}{48807504148}a^{10}+\frac{2815818991}{174624344}a^{6}-\frac{801662383}{24403752074}a^{5}-\frac{178378536}{12201876037}a^{4}-\frac{21878013}{21828043}$, $\frac{3385769929}{799662147960832}a^{35}-\frac{1041029}{199915536990208}a^{34}-\frac{31257627}{18596794138624}a^{33}-\frac{119669137}{199915536990208}a^{32}+\frac{169635}{715261313024}a^{30}-\frac{128659072869}{99957768495104}a^{29}+\frac{72050165}{49978884247552}a^{28}+\frac{51064467355}{99957768495104}a^{27}+\frac{2273649231}{12494721061888}a^{26}-\frac{3222343}{44703832064}a^{24}+\frac{8904562148565}{24989442123776}a^{23}-\frac{9877905599}{24989442123776}a^{22}-\frac{1767066588773}{12494721061888}a^{21}-\frac{157360459935}{3123680265472}a^{20}+\frac{223020055}{11175958016}a^{18}-\frac{4147513593503}{390460033184}a^{17}+\frac{909694973}{780920066368}a^{16}+\frac{6572280749065}{1561840132736}a^{15}+\frac{2344901835491}{1561840132736}a^{14}-\frac{829482275}{1396994752}a^{12}+\frac{112427715097053}{390460033184}a^{11}-\frac{164373}{2270116472}a^{10}-\frac{11133846054613}{97615008296}a^{9}-\frac{1986810430647}{48807504148}a^{8}+\frac{2815818991}{174624344}a^{6}-\frac{20314590453}{1135058236}a^{5}-\frac{120915407855}{12201876037}a^{4}+\frac{197964971}{12201876037}a^{3}+\frac{717994494}{283764559}a^{2}-\frac{21878013}{21828043}$, $\frac{9785137235}{15\!\cdots\!64}a^{35}+\frac{2726975}{799662147960832}a^{34}-\frac{479047}{399831073980416}a^{32}+\frac{8949039}{9298397069312}a^{31}-\frac{307833957}{399831073980416}a^{30}-\frac{46479438693}{24989442123776}a^{29}-\frac{188735375}{199915536990208}a^{28}+\frac{33155095}{99957768495104}a^{26}-\frac{14619769469}{49978884247552}a^{25}+\frac{2924387261}{12494721061888}a^{24}+\frac{3216866414805}{6247360530944}a^{23}+\frac{6468625301}{24989442123776}a^{22}-\frac{568213383}{6247360530944}a^{20}+\frac{505910055761}{6247360530944}a^{19}-\frac{202398381485}{3123680265472}a^{18}-\frac{47947522386795}{3123680265472}a^{17}-\frac{2382945575}{3123680265472}a^{16}+\frac{418611439}{1561840132736}a^{14}-\frac{1881639855205}{780920066368}a^{13}+\frac{1508202792225}{780920066368}a^{12}+\frac{40615690558941}{97615008296}a^{11}+\frac{430575}{9080465888}a^{10}-\frac{75639}{4540232944}a^{8}+\frac{796650269887}{12201876037}a^{7}-\frac{2555452720757}{48807504148}a^{6}-\frac{7338858741}{283764559}a^{5}+\frac{311044791177}{48807504148}a^{4}-\frac{30803119380}{12201876037}a^{2}-\frac{113354494}{12201876037}a+\frac{923490714}{283764559}$, $\frac{3520629}{2324599267328}a^{35}+\frac{3520629}{2324599267328}a^{34}+\frac{119669137}{199915536990208}a^{32}+\frac{169635}{715261313024}a^{31}+\frac{18809339}{399831073980416}a^{30}-\frac{92024515237}{199915536990208}a^{29}-\frac{92024515237}{199915536990208}a^{28}-\frac{2273649231}{12494721061888}a^{26}-\frac{3222343}{44703832064}a^{25}-\frac{22297317}{1561840132736}a^{24}+\frac{199029372171}{1561840132736}a^{23}+\frac{199029372171}{1561840132736}a^{22}+\frac{157360459935}{3123680265472}a^{20}+\frac{223020055}{11175958016}a^{19}+\frac{1543209045}{390460033184}a^{18}-\frac{740253321255}{195230016592}a^{17}-\frac{740253321255}{195230016592}a^{16}-\frac{2344901835491}{1561840132736}a^{14}-\frac{829482275}{1396994752}a^{13}-\frac{11332767523}{97615008296}a^{12}+\frac{5015625515159}{48807504148}a^{11}+\frac{5015625515159}{48807504148}a^{10}+\frac{1986810430647}{48807504148}a^{8}+\frac{2815818991}{174624344}a^{7}+\frac{38968668858}{12201876037}a^{6}-\frac{178378536}{12201876037}a^{5}-\frac{178378536}{12201876037}a^{4}-\frac{717994494}{283764559}a^{2}-\frac{21878013}{21828043}a-\frac{56330064}{283764559}$, $\frac{2726975}{799662147960832}a^{34}+\frac{1437141}{399831073980416}a^{33}+\frac{313177}{199915536990208}a^{32}+\frac{147307}{199915536990208}a^{31}-\frac{188735375}{199915536990208}a^{28}-\frac{99465285}{99957768495104}a^{27}-\frac{21675145}{49978884247552}a^{26}-\frac{10195195}{49978884247552}a^{25}+\frac{6468625301}{24989442123776}a^{22}+\frac{1704640149}{6247360530944}a^{21}+\frac{1486362061}{12494721061888}a^{20}+\frac{349935295}{6247360530944}a^{19}-\frac{2382945575}{3123680265472}a^{16}-\frac{1255834317}{1561840132736}a^{15}-\frac{273667249}{780920066368}a^{14}-\frac{128723059}{780920066368}a^{13}+\frac{430575}{9080465888}a^{10}+\frac{226917}{4540232944}a^{9}+\frac{49449}{2270116472}a^{8}+\frac{23259}{2270116472}a^{7}+\frac{311044791177}{48807504148}a^{4}+\frac{172616840243}{24403752074}a^{3}+\frac{95795128069}{24403752074}a^{2}+\frac{34188889309}{12201876037}a+1$, $\frac{3092595}{9298397069312}a^{33}-\frac{52476131}{199915536990208}a^{30}-\frac{10104419473}{99957768495104}a^{27}+\frac{306595747}{3844529557504}a^{24}+\frac{174831611405}{6247360530944}a^{21}-\frac{34483228601}{1561840132736}a^{18}-\frac{650254179025}{780920066368}a^{15}+\frac{39350284897}{60070774336}a^{12}+\frac{1102595024295}{48807504148}a^{9}-\frac{1747031690793}{97615008296}a^{6}-\frac{39172870}{12201876037}a^{3}+\frac{523257490}{938605849}$, $\frac{78917949}{18596794138624}a^{35}+\frac{1041029}{199915536990208}a^{34}-\frac{1341203679}{799662147960832}a^{33}-\frac{5577147}{9298397069312}a^{32}-\frac{8057849899}{6247360530944}a^{29}-\frac{72050165}{49978884247552}a^{28}+\frac{25482501035}{49978884247552}a^{27}+\frac{18222348943}{99957768495104}a^{26}+\frac{4461415798851}{12494721061888}a^{23}+\frac{9877905599}{24989442123776}a^{22}-\frac{1763657308475}{12494721061888}a^{21}-\frac{315289133253}{6247360530944}a^{20}-\frac{16593419486655}{1561840132736}a^{17}-\frac{909694973}{780920066368}a^{16}+\frac{1642756228687}{390460033184}a^{15}+\frac{1172660223465}{780920066368}a^{14}+\frac{112427767388751}{390460033184}a^{11}+\frac{164373}{2270116472}a^{10}-\frac{22267682351795}{195230016592}a^{9}-\frac{7947244975065}{195230016592}a^{8}-\frac{499813677}{12201876037}a^{5}+\frac{120915407855}{12201876037}a^{4}+\frac{4023552795}{567529118}a^{3}+\frac{70643862}{12201876037}a^{2}-1$
| 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: | \( 445456911908259.2 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 445456911908259.2 \cdot 324}{18\cdot\sqrt{200687573080369568029416132506181048520658333428355416190877696}}\cr\approx \mathstrut & 0.131842550855242 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 - 304*x^30 + 84160*x^24 - 2508800*x^18 + 68005888*x^12 - 4227072*x^6 + 262144)
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 - 304*x^30 + 84160*x^24 - 2508800*x^18 + 68005888*x^12 - 4227072*x^6 + 262144, 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 - 304*x^30 + 84160*x^24 - 2508800*x^18 + 68005888*x^12 - 4227072*x^6 + 262144);
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 - 304*x^30 + 84160*x^24 - 2508800*x^18 + 68005888*x^12 - 4227072*x^6 + 262144);
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
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 $C_6^2$ |
Intermediate fields
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 | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.3.0.1}{3} }^{12}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.3.0.1}{3} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
| 2.12.18.15 | $x^{12} - 12 x^{11} - 56 x^{9} + 2372 x^{8} + 8992 x^{7} + 115648 x^{6} + 164160 x^{5} + 1305648 x^{4} - 282560 x^{3} + 5591296 x^{2} - 9640320 x + 8497856$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(3\)
| Deg $18$ | $6$ | $3$ | $27$ | |||
| Deg $18$ | $6$ | $3$ | $27$ | ||||
|
\(7\)
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |