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$$\chi_{504}(1,·)$, $\chi_{504}(389,·)$, $\chi_{504}(193,·)$, $\chi_{504}(137,·)$, $\chi_{504}(109,·)$, $\chi_{504}(401,·)$, $\chi_{504}(277,·)$, $\chi_{504}(25,·)$, $\chi_{504}(281,·)$, $\chi_{504}(29,·)$, $\chi_{504}(289,·)$, $\chi_{504}(37,·)$, $\chi_{504}(65,·)$, $\chi_{504}(169,·)$, $\chi_{504}(305,·)$, $\chi_{504}(53,·)$, $\chi_{504}(317,·)$, $\chi_{504}(449,·)$, $\chi_{504}(197,·)$, $\chi_{504}(457,·)$, $\chi_{504}(205,·)$, $\chi_{504}(337,·)$, $\chi_{504}(85,·)$, $\chi_{504}(473,·)$, $\chi_{504}(221,·)$, $\chi_{504}(421,·)$, $\chi_{504}(485,·)$, $\chi_{504}(361,·)$, $\chi_{504}(365,·)$, $\chi_{504}(445,·)$, $\chi_{504}(113,·)$, $\chi_{504}(373,·)$, $\chi_{504}(233,·)$, $\chi_{504}(121,·)$, $\chi_{504}(253,·)$, $\chi_{504}(149,·)$$\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
$C_{18}\times C_{54}$, which has order $972$ (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: |
\( -\frac{3520629}{2324599267328} a^{34} - \frac{92024515237}{199915536990208} a^{28} - \frac{199029372171}{1561840132736} a^{22} - \frac{740253321255}{195230016592} a^{16} - \frac{5015625515159}{48807504148} a^{10} - \frac{178378536}{12201876037} a^{4} \)
(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: |
$\frac{989169}{9298397069312}a^{30}+\frac{1616089471}{49978884247552}a^{24}+\frac{55920031631}{6247360530944}a^{18}+\frac{207984322555}{780920066368}a^{12}+\frac{350263798337}{48807504148}a^{6}+\frac{12529474}{12201876037}$, $\frac{38008313}{199915536990208}a^{30}+\frac{1444532665}{24989442123776}a^{24}+\frac{99976866025}{6247360530944}a^{18}+\frac{373018451541}{780920066368}a^{12}+\frac{1262293465105}{97615008296}a^{6}+\frac{228084105}{283764559}$, $\frac{3520629}{2324599267328}a^{34}-\frac{119669137}{199915536990208}a^{32}+\frac{169635}{715261313024}a^{30}+\frac{92024515237}{199915536990208}a^{28}-\frac{2273649231}{12494721061888}a^{26}+\frac{3222343}{44703832064}a^{24}+\frac{199029372171}{1561840132736}a^{22}-\frac{157360459935}{3123680265472}a^{20}+\frac{223020055}{11175958016}a^{18}+\frac{740253321255}{195230016592}a^{16}-\frac{2344901835491}{1561840132736}a^{14}+\frac{829482275}{1396994752}a^{12}+\frac{5015625515159}{48807504148}a^{10}-\frac{1986810430647}{48807504148}a^{8}+\frac{2815818991}{174624344}a^{6}+\frac{178378536}{12201876037}a^{4}-\frac{717994494}{283764559}a^{2}+\frac{21878013}{21828043}$, $\frac{119669137}{199915536990208}a^{32}+\frac{2273649231}{12494721061888}a^{26}+\frac{157360459935}{3123680265472}a^{20}+\frac{2344901835491}{1561840132736}a^{14}+\frac{1986810430647}{48807504148}a^{8}+\frac{717994494}{283764559}a^{2}-1$, $\frac{3520629}{2324599267328}a^{34}-\frac{579}{429457408}a^{32}+\frac{26625999}{49978884247552}a^{30}+\frac{92024515237}{199915536990208}a^{28}-\frac{472947}{1154166784}a^{26}+\frac{4047484785}{24989442123776}a^{24}+\frac{199029372171}{1561840132736}a^{22}-\frac{32732221}{288541696}a^{20}+\frac{280128552225}{6247360530944}a^{18}+\frac{740253321255}{195230016592}a^{16}-\frac{121741505}{36067712}a^{14}+\frac{261130550125}{195230016592}a^{12}+\frac{5015625515159}{48807504148}a^{10}-\frac{206190241}{2254232}a^{8}+\frac{3536862625545}{97615008296}a^{6}+\frac{178378536}{12201876037}a^{4}-\frac{3667}{281779}a^{2}+\frac{639076545}{283764559}$, $\frac{3013597377}{15\!\cdots\!64}a^{34}+\frac{119669137}{199915536990208}a^{32}+\frac{57258681903}{99957768495104}a^{28}+\frac{2273649231}{12494721061888}a^{26}+\frac{3962903510655}{24989442123776}a^{22}+\frac{157360459935}{3123680265472}a^{20}+\frac{14767413638771}{3123680265472}a^{16}+\frac{2344901835491}{1561840132736}a^{14}+\frac{50035047138711}{390460033184}a^{10}+\frac{1986810430647}{48807504148}a^{8}+\frac{9040844511}{1135058236}a^{4}+\frac{717994494}{283764559}a^{2}-1$, $\frac{2726975}{799662147960832}a^{34}+\frac{119669137}{199915536990208}a^{32}+\frac{147307}{199915536990208}a^{30}+\frac{188735375}{199915536990208}a^{28}+\frac{2273649231}{12494721061888}a^{26}+\frac{10195195}{49978884247552}a^{24}+\frac{6468625301}{24989442123776}a^{22}+\frac{157360459935}{3123680265472}a^{20}+\frac{349935295}{6247360530944}a^{18}+\frac{2382945575}{3123680265472}a^{16}+\frac{2344901835491}{1561840132736}a^{14}+\frac{128723059}{780920066368}a^{12}+\frac{430575}{9080465888}a^{10}+\frac{1986810430647}{48807504148}a^{8}+\frac{23259}{2270116472}a^{6}-\frac{311044791177}{48807504148}a^{4}+\frac{717994494}{283764559}a^{2}-\frac{21987013272}{12201876037}$, $\frac{1208369401}{799662147960832}a^{34}-\frac{596858575}{799662147960832}a^{32}+\frac{169635}{715261313024}a^{30}+\frac{45917889931}{99957768495104}a^{28}-\frac{2835197993}{12494721061888}a^{26}+\frac{3222343}{44703832064}a^{24}+\frac{3178001329435}{24989442123776}a^{22}-\frac{196225545305}{3123680265472}a^{20}+\frac{223020055}{11175958016}a^{18}+\frac{11841670194505}{3123680265472}a^{16}-\frac{1462871722133}{780920066368}a^{14}+\frac{829482275}{1396994752}a^{12}+\frac{40124985606547}{390460033184}a^{10}-\frac{2477515383041}{48807504148}a^{8}+\frac{2815818991}{174624344}a^{6}+\frac{7250193147}{1135058236}a^{4}-\frac{895325682}{283764559}a^{2}+\frac{21878013}{21828043}$, $\frac{6264737}{799662147960832}a^{35}-\frac{119669137}{199915536990208}a^{32}+\frac{433585745}{199915536990208}a^{29}-\frac{2273649231}{12494721061888}a^{26}+\frac{14860168839}{24989442123776}a^{23}-\frac{157360459935}{3123680265472}a^{20}+\frac{5474390969}{3123680265472}a^{17}-\frac{2344901835491}{1561840132736}a^{14}+\frac{989169}{9080465888}a^{11}-\frac{1986810430647}{48807504148}a^{8}-\frac{174727823632}{12201876037}a^{5}-\frac{717994494}{283764559}a^{2}$, $\frac{5577147}{9298397069312}a^{32}-\frac{82935}{49978884247552}a^{31}-\frac{169635}{715261313024}a^{30}+\frac{18222348943}{99957768495104}a^{26}-\frac{5739975}{12494721061888}a^{25}-\frac{3222343}{44703832064}a^{24}+\frac{315289133253}{6247360530944}a^{20}-\frac{786491471}{6247360530944}a^{19}-\frac{223020055}{11175958016}a^{18}+\frac{1172660223465}{780920066368}a^{14}-\frac{72472095}{195230016592}a^{13}-\frac{829482275}{1396994752}a^{12}+\frac{7947244975065}{195230016592}a^{8}-\frac{13095}{567529118}a^{7}-\frac{2815818991}{174624344}a^{6}+\frac{70643862}{12201876037}a^{2}+\frac{27417349451}{12201876037}a-\frac{21878013}{21828043}$, $\frac{5577147}{9298397069312}a^{32}-\frac{26625999}{49978884247552}a^{31}+\frac{18222348943}{99957768495104}a^{26}-\frac{4047484785}{24989442123776}a^{25}+\frac{315289133253}{6247360530944}a^{20}-\frac{280128552225}{6247360530944}a^{19}+\frac{1172660223465}{780920066368}a^{14}-\frac{261130550125}{195230016592}a^{13}+\frac{7947244975065}{195230016592}a^{8}-\frac{3536862625545}{97615008296}a^{7}+\frac{70643862}{12201876037}a^{2}-\frac{639076545}{283764559}a+1$, $\frac{579}{429457408}a^{33}-\frac{147307}{199915536990208}a^{30}+\frac{472947}{1154166784}a^{27}-\frac{10195195}{49978884247552}a^{24}+\frac{32732221}{288541696}a^{21}-\frac{349935295}{6247360530944}a^{18}+\frac{121741505}{36067712}a^{15}-\frac{128723059}{780920066368}a^{12}+\frac{206190241}{2254232}a^{9}-\frac{23259}{2270116472}a^{6}+\frac{3667}{281779}a^{3}+\frac{21987013272}{12201876037}$, $\frac{126578271}{37193588277248}a^{35}+\frac{810787}{399831073980416}a^{32}+\frac{206786729413}{199915536990208}a^{29}+\frac{56114995}{99957768495104}a^{26}+\frac{7155765008929}{24989442123776}a^{23}+\frac{1922918237}{12494721061888}a^{20}+\frac{26614558224245}{3123680265472}a^{17}+\frac{708499819}{1561840132736}a^{14}+\frac{90160075279525}{390460033184}a^{11}+\frac{128019}{4540232944}a^{8}+\frac{801662383}{24403752074}a^{5}-\frac{38410856087}{12201876037}a^{2}$, $\frac{4974903}{799662147960832}a^{35}+\frac{1207079567}{799662147960832}a^{34}-\frac{969031127}{399831073980416}a^{33}+\frac{1554211671}{799662147960832}a^{32}-\frac{383850583}{399831073980416}a^{31}+\frac{399303}{1162299633664}a^{30}+\frac{344315655}{199915536990208}a^{29}+\frac{22936627443}{49978884247552}a^{28}-\frac{36823293007}{49978884247552}a^{27}+\frac{7382496455}{12494721061888}a^{26}-\frac{7293307187}{24989442123776}a^{25}+\frac{5218668945}{49978884247552}a^{24}+\frac{2950205959}{6247360530944}a^{23}+\frac{198433874027}{1561840132736}a^{22}-\frac{2548559489695}{12494721061888}a^{21}+\frac{510946465175}{3123680265472}a^{20}-\frac{504773628995}{6247360530944}a^{19}+\frac{22573530297}{780920066368}a^{18}+\frac{4347279711}{3123680265472}a^{17}+\frac{11840543083247}{3123680265472}a^{16}-\frac{4748384179507}{780920066368}a^{15}+\frac{475971694703}{97615008296}a^{14}-\frac{940610621883}{390460033184}a^{13}+\frac{83958114285}{97615008296}a^{12}+\frac{785511}{9080465888}a^{11}+\frac{40124976849253}{390460033184}a^{10}-\frac{32177743883959}{195230016592}a^{9}+\frac{6451136244335}{48807504148}a^{8}-\frac{6373198906619}{97615008296}a^{7}+\frac{2274570412643}{97615008296}a^{6}-\frac{280241671797}{24403752074}a^{5}+\frac{450186256255}{48807504148}a^{4}-\frac{5814204159}{567529118}a^{3}+\frac{2331314670}{283764559}a^{2}-\frac{1151574819}{283764559}a+\frac{12242338741}{12201876037}$, $\frac{9785137235}{15\!\cdots\!64}a^{35}-\frac{182908335}{37193588277248}a^{34}+\frac{280991}{49978884247552}a^{33}+\frac{119669137}{199915536990208}a^{32}-\frac{169635}{715261313024}a^{30}+\frac{46479438693}{24989442123776}a^{29}-\frac{149405622325}{99957768495104}a^{28}+\frac{19447535}{12494721061888}a^{27}+\frac{2273649231}{12494721061888}a^{26}-\frac{3222343}{44703832064}a^{24}+\frac{3216866414805}{6247360530944}a^{23}-\frac{10340234963665}{24989442123776}a^{22}+\frac{5332198535}{12494721061888}a^{21}+\frac{157360459935}{3123680265472}a^{20}-\frac{223020055}{11175958016}a^{18}+\frac{47947522386795}{3123680265472}a^{17}-\frac{38458611364325}{3123680265472}a^{16}+\frac{245541767}{195230016592}a^{15}+\frac{2344901835491}{1561840132736}a^{14}-\frac{829482275}{1396994752}a^{12}+\frac{40615690558941}{97615008296}a^{11}-\frac{130285079400797}{390460033184}a^{10}+\frac{44367}{567529118}a^{9}+\frac{1986810430647}{48807504148}a^{8}-\frac{2815818991}{174624344}a^{6}+\frac{7338858741}{283764559}a^{5}-\frac{1158419455}{24403752074}a^{4}-\frac{249438552417}{24403752074}a^{3}+\frac{717994494}{283764559}a^{2}-\frac{49970}{21828043}$, $\frac{604040687}{123024945840128}a^{35}-\frac{76433397}{37193588277248}a^{34}+\frac{241107155}{399831073980416}a^{33}-\frac{134271419}{399831073980416}a^{32}+\frac{82935}{24989442123776}a^{31}+\frac{25851235}{199915536990208}a^{30}+\frac{298377658905}{199915536990208}a^{29}-\frac{124866633649}{199915536990208}a^{28}+\frac{18311619033}{99957768495104}a^{27}-\frac{10193689563}{99957768495104}a^{26}+\frac{5739975}{6247360530944}a^{25}+\frac{1966099613}{49978884247552}a^{24}+\frac{397129799801}{961132389376}a^{23}-\frac{4320958277003}{24989442123776}a^{22}+\frac{633637611509}{12494721061888}a^{21}-\frac{352722567813}{12494721061888}a^{20}+\frac{786491471}{3123680265472}a^{19}+\frac{17012077131}{1561840132736}a^{18}+\frac{9613284243339}{780920066368}a^{17}-\frac{16071013442215}{3123680265472}a^{16}+\frac{586611889547}{390460033184}a^{15}-\frac{325408867327}{390460033184}a^{14}+\frac{72472095}{97615008296}a^{13}+\frac{63859705763}{195230016592}a^{12}+\frac{5010962956405}{15017693584}a^{11}-\frac{54445451255433}{390460033184}a^{10}+\frac{7947253732359}{195230016592}a^{9}-\frac{2205194427237}{97615008296}a^{8}+\frac{13095}{283764559}a^{7}+\frac{873513219021}{97615008296}a^{6}+\frac{350614066719}{24403752074}a^{5}-\frac{484078181}{24403752074}a^{4}-\frac{69143331605}{12201876037}a^{3}+\frac{126147729157}{24403752074}a^{2}-\frac{54834698902}{12201876037}a+\frac{31787554263}{12201876037}$, $\frac{24985}{1430522626048}a^{35}+\frac{1}{131072}a^{34}-\frac{50752917}{9298397069312}a^{33}+\frac{2726975}{399831073980416}a^{32}+\frac{119669137}{99957768495104}a^{31}+\frac{1729225}{357630656512}a^{29}+\frac{19}{8192}a^{28}-\frac{165826681531}{99957768495104}a^{27}+\frac{188735375}{99957768495104}a^{26}+\frac{2273649231}{6247360530944}a^{25}+\frac{7408233}{5587979008}a^{23}+\frac{1315}{2048}a^{22}-\frac{2869180821483}{6247360530944}a^{21}+\frac{6468625301}{12494721061888}a^{20}+\frac{157360459935}{1561840132736}a^{19}+\frac{21832945}{5587979008}a^{17}+\frac{1225}{64}a^{16}-\frac{10671392916615}{780920066368}a^{15}+\frac{2382945575}{1561840132736}a^{14}+\frac{2344901835491}{780920066368}a^{13}+\frac{169635}{698497376}a^{11}+\frac{16603}{32}a^{10}-\frac{72302763267479}{195230016592}a^{9}+\frac{430575}{4540232944}a^{8}+\frac{1986810430647}{24403752074}a^{7}-\frac{2809372861}{87312172}a^{5}+\frac{129}{4}a^{4}-\frac{642870282}{12201876037}a^{3}-\frac{311044791177}{24403752074}a^{2}+\frac{1435988988}{283764559}a+1$
| 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: | \( 160258501280890.78 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 160258501280890.78 \cdot 972}{18\cdot\sqrt{200687573080369568029416132506181048520658333428355416190877696}}\cr\approx \mathstrut & 0.142295847523826 \end{aligned}\] (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 | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.3.0.1}{3} }^{12}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ |
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
| 2.12.18.23 | $x^{12} - 12 x^{11} + 48 x^{10} - 344 x^{9} + 8244 x^{8} - 31136 x^{7} + 54848 x^{6} - 23104 x^{5} + 18864 x^{4} - 7360 x^{3} + 5120 x^{2} + 5760 x + 1472$ | $2$ | $6$ | $18$ | $C_6\times C_2$ | $[3]^{6}$ | |
|
\(3\)
| Deg $36$ | $6$ | $6$ | $54$ | |||
|
\(7\)
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
| 7.9.6.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 249 x^{6} + 396 x^{5} + 1944 x^{4} + 2631 x^{3} - 2358 x^{2} - 756 x + 11915$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ |