Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^30 + 44*x^28 - 208*x^26 + 911*x^24 - 2776*x^22 + 7272*x^20 - 17024*x^18 + 34257*x^16 - 47152*x^14 + 59688*x^12 - 65552*x^10 + 50831*x^8 - 3464*x^6 + 236*x^4 - 16*x^2 + 1)
gp: K = bnfinit(y^32 - 8*y^30 + 44*y^28 - 208*y^26 + 911*y^24 - 2776*y^22 + 7272*y^20 - 17024*y^18 + 34257*y^16 - 47152*y^14 + 59688*y^12 - 65552*y^10 + 50831*y^8 - 3464*y^6 + 236*y^4 - 16*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 8*x^30 + 44*x^28 - 208*x^26 + 911*x^24 - 2776*x^22 + 7272*x^20 - 17024*x^18 + 34257*x^16 - 47152*x^14 + 59688*x^12 - 65552*x^10 + 50831*x^8 - 3464*x^6 + 236*x^4 - 16*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 8*x^30 + 44*x^28 - 208*x^26 + 911*x^24 - 2776*x^22 + 7272*x^20 - 17024*x^18 + 34257*x^16 - 47152*x^14 + 59688*x^12 - 65552*x^10 + 50831*x^8 - 3464*x^6 + 236*x^4 - 16*x^2 + 1)
\( x^{32} - 8 x^{30} + 44 x^{28} - 208 x^{26} + 911 x^{24} - 2776 x^{22} + 7272 x^{20} - 17024 x^{18} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(203282392447840896882957090816000000000000000000000000\)
\(\medspace = 2^{96}\cdot 3^{16}\cdot 5^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(46.33\) | 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: | $2^{3}3^{1/2}5^{3/4}\approx 46.331687411530766$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $32$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(240=2^{4}\cdot 3\cdot 5\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(139,·)$, $\chi_{240}(143,·)$, $\chi_{240}(19,·)$, $\chi_{240}(149,·)$, $\chi_{240}(23,·)$, $\chi_{240}(29,·)$, $\chi_{240}(163,·)$, $\chi_{240}(167,·)$, $\chi_{240}(169,·)$, $\chi_{240}(43,·)$, $\chi_{240}(173,·)$, $\chi_{240}(47,·)$, $\chi_{240}(49,·)$, $\chi_{240}(53,·)$, $\chi_{240}(187,·)$, $\chi_{240}(191,·)$, $\chi_{240}(193,·)$, $\chi_{240}(67,·)$, $\chi_{240}(197,·)$, $\chi_{240}(71,·)$, $\chi_{240}(73,·)$, $\chi_{240}(77,·)$, $\chi_{240}(211,·)$, $\chi_{240}(217,·)$, $\chi_{240}(91,·)$, $\chi_{240}(221,·)$, $\chi_{240}(97,·)$, $\chi_{240}(101,·)$, $\chi_{240}(239,·)$, $\chi_{240}(119,·)$, $\chi_{240}(121,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{1041753338401}a^{26}-\frac{79843284091}{1041753338401}a^{24}+\frac{486571795735}{1041753338401}a^{22}-\frac{212202007274}{1041753338401}a^{20}-\frac{505807280647}{1041753338401}a^{18}+\frac{402287136637}{1041753338401}a^{16}+\frac{88417645693}{1041753338401}a^{14}-\frac{432555802638}{1041753338401}a^{12}-\frac{71680044697}{1041753338401}a^{10}+\frac{115313573804}{1041753338401}a^{8}+\frac{285663179810}{1041753338401}a^{6}+\frac{421135329286}{1041753338401}a^{4}+\frac{96513802723}{1041753338401}a^{2}-\frac{337948059055}{1041753338401}$, $\frac{1}{1041753338401}a^{27}-\frac{79843284091}{1041753338401}a^{25}+\frac{486571795735}{1041753338401}a^{23}-\frac{212202007274}{1041753338401}a^{21}-\frac{505807280647}{1041753338401}a^{19}+\frac{402287136637}{1041753338401}a^{17}+\frac{88417645693}{1041753338401}a^{15}-\frac{432555802638}{1041753338401}a^{13}-\frac{71680044697}{1041753338401}a^{11}+\frac{115313573804}{1041753338401}a^{9}+\frac{285663179810}{1041753338401}a^{7}+\frac{421135329286}{1041753338401}a^{5}+\frac{96513802723}{1041753338401}a^{3}-\frac{337948059055}{1041753338401}a$, $\frac{1}{1041753338401}a^{28}+\frac{25054628700}{1041753338401}a^{24}-\frac{110659118192}{1041753338401}a^{22}+\frac{384180371535}{1041753338401}a^{20}-\frac{402287135589}{1041753338401}a^{18}-\frac{16052180812}{1041753338401}a^{16}+\frac{323269475077}{1041753338401}a^{14}+\frac{270132982402}{1041753338401}a^{12}+\frac{366695375033}{1041753338401}a^{10}+\frac{284680379083}{1041753338401}a^{8}+\frac{119244722731}{1041753338401}a^{6}-\frac{282094818063}{1041753338401}a^{4}-\frac{334503429196}{1041753338401}a^{2}-\frac{125192178357}{1041753338401}$, $\frac{1}{1041753338401}a^{29}+\frac{25054628700}{1041753338401}a^{25}-\frac{110659118192}{1041753338401}a^{23}+\frac{384180371535}{1041753338401}a^{21}-\frac{402287135589}{1041753338401}a^{19}-\frac{16052180812}{1041753338401}a^{17}+\frac{323269475077}{1041753338401}a^{15}+\frac{270132982402}{1041753338401}a^{13}+\frac{366695375033}{1041753338401}a^{11}+\frac{284680379083}{1041753338401}a^{9}+\frac{119244722731}{1041753338401}a^{7}-\frac{282094818063}{1041753338401}a^{5}-\frac{334503429196}{1041753338401}a^{3}-\frac{125192178357}{1041753338401}a$, $\frac{1}{1041753338401}a^{30}-\frac{182905806350}{1041753338401}a^{20}+\frac{234369048594}{1041753338401}a^{10}+\frac{415161683759}{1041753338401}$, $\frac{1}{1041753338401}a^{31}-\frac{182905806350}{1041753338401}a^{21}+\frac{234369048594}{1041753338401}a^{11}+\frac{415161683759}{1041753338401}a$
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
$C_{2}\times C_{2}\times C_{20}$, which has order $80$ (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: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{6579480000}{1041753338401} a^{30} + \frac{47372256000}{1041753338401} a^{28} - \frac{247717422000}{1041753338401} a^{26} + \frac{1139562526801}{1041753338401} a^{24} - \frac{4913555664000}{1041753338401} a^{22} + \frac{13537938048000}{1041753338401} a^{20} - \frac{33533964690000}{1041753338401} a^{18} + \frac{74645516496000}{1041753338401} a^{16} - \frac{138181955344920}{1041753338401} a^{14} + \frac{135521497248000}{1041753338401} a^{12} - \frac{155797151790000}{1041753338401} a^{10} + \frac{132637053216000}{1041753338401} a^{8} - \frac{9038889624000}{1041753338401} a^{6} - \frac{223697106722820}{1041753338401} a^{4} - \frac{41779698000}{1041753338401} a^{2} + \frac{2631792000}{1041753338401} \)
(order $10$)
| 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: |
$\frac{77572923512}{1041753338401}a^{30}-\frac{615317770412}{1041753338401}a^{28}+\frac{3371428936528}{1041753338401}a^{26}-\frac{15906198777297}{1041753338401}a^{24}+\frac{69588582703432}{1041753338401}a^{22}-\frac{210615737237312}{1041753338401}a^{20}+\frac{549800427630528}{1041753338401}a^{18}-\frac{12\!\cdots\!88}{1041753338401}a^{16}+\frac{25\!\cdots\!04}{1041753338401}a^{14}-\frac{34\!\cdots\!24}{1041753338401}a^{12}+\frac{43\!\cdots\!56}{1041753338401}a^{10}-\frac{47\!\cdots\!57}{1041753338401}a^{8}+\frac{36\!\cdots\!72}{1041753338401}a^{6}-\frac{22224181602748}{1041753338401}a^{4}+\frac{16796232366832}{1041753338401}a^{2}-\frac{1138526888192}{1041753338401}$, $\frac{2033684}{1041753338401}a^{30}+\frac{2141721264}{1041753338401}a^{20}+\frac{221055569167}{1041753338401}a^{10}+\frac{2012513233290}{1041753338401}$, $\frac{438677800}{1041753338401}a^{31}-\frac{2412727900}{1041753338401}a^{29}+\frac{11405622800}{1041753338401}a^{27}-\frac{49954434475}{1041753338401}a^{25}+\frac{209671852455}{1041753338401}a^{23}-\frac{398758120200}{1041753338401}a^{21}+\frac{933506358400}{1041753338401}a^{19}-\frac{1878473174325}{1041753338401}a^{17}+\frac{2585566953200}{1041753338401}a^{15}+\frac{2332111453182}{1041753338401}a^{13}+\frac{3594525893200}{1041753338401}a^{11}-\frac{2787303906475}{1041753338401}a^{9}+\frac{189947487400}{1041753338401}a^{7}-\frac{12940995100}{1041753338401}a^{5}+\frac{36905889653263}{1041753338401}a^{3}-\frac{54834725}{1041753338401}a$, $\frac{171204088}{1041753338401}a^{31}-\frac{941622484}{1041753338401}a^{29}+\frac{4451306288}{1041753338401}a^{27}-\frac{19495865521}{1041753338401}a^{25}+\frac{81838660680}{1041753338401}a^{23}-\frac{155624515992}{1041753338401}a^{21}+\frac{364322299264}{1041753338401}a^{19}-\frac{733117305327}{1041753338401}a^{17}+\frac{1009076894672}{1041753338401}a^{15}+\frac{919609660376}{1041753338401}a^{13}+\frac{1402846297072}{1041753338401}a^{11}-\frac{1087809374641}{1041753338401}a^{9}+\frac{74131370104}{1041753338401}a^{7}-\frac{5050520596}{1041753338401}a^{5}+\frac{15286898838064}{1041753338401}a^{3}-\frac{21400511}{1041753338401}a$, $\frac{6312006288}{1041753338401}a^{30}-\frac{45901150584}{1041753338401}a^{28}+\frac{240763105488}{1041753338401}a^{26}-\frac{1109103957847}{1041753338401}a^{24}+\frac{4785722472225}{1041753338401}a^{22}-\frac{13294804443792}{1041753338401}a^{20}+\frac{32964780630864}{1041753338401}a^{18}-\frac{73500160627002}{1041753338401}a^{16}+\frac{136605465286392}{1041753338401}a^{14}-\frac{136933999040806}{1041753338401}a^{12}+\frac{153605472193872}{1041753338401}a^{10}-\frac{130937558684166}{1041753338401}a^{8}+\frac{8923073506704}{1041753338401}a^{6}+\frac{223704997197324}{1041753338401}a^{4}-\frac{21577211117199}{1041753338401}a^{2}+\frac{1039154980615}{1041753338401}$, $\frac{143557555472}{1041753338401}a^{30}-\frac{1143623637924}{1041753338401}a^{28}+\frac{6277834416352}{1041753338401}a^{26}-\frac{29647116809760}{1041753338401}a^{24}+\frac{129774710682480}{1041753338401}a^{22}-\frac{394108713590568}{1041753338401}a^{20}+\frac{10\!\cdots\!77}{1041753338401}a^{18}-\frac{24\!\cdots\!13}{1041753338401}a^{16}+\frac{48\!\cdots\!68}{1041753338401}a^{14}-\frac{66\!\cdots\!96}{1041753338401}a^{12}+\frac{83\!\cdots\!08}{1041753338401}a^{10}-\frac{91\!\cdots\!97}{1041753338401}a^{8}+\frac{69\!\cdots\!56}{1041753338401}a^{6}-\frac{251382947152424}{1041753338401}a^{4}+\frac{17122110836096}{1041753338401}a^{2}-\frac{2196998864449}{1041753338401}$, $\frac{6483210376}{1041753338401}a^{30}-\frac{46842773068}{1041753338401}a^{28}+\frac{245214411776}{1041753338401}a^{26}-\frac{1128599823368}{1041753338401}a^{24}+\frac{68557199055}{14672582231}a^{22}-\frac{13450428959784}{1041753338401}a^{20}+\frac{33329102930128}{1041753338401}a^{18}-\frac{74233277932329}{1041753338401}a^{16}+\frac{137614542181064}{1041753338401}a^{14}-\frac{136014389380430}{1041753338401}a^{12}+\frac{155008318490944}{1041753338401}a^{10}-\frac{132025368058807}{1041753338401}a^{8}+\frac{126721195448}{14672582231}a^{6}+\frac{223699946676728}{1041753338401}a^{4}-\frac{92810957184}{13186751119}a^{2}+\frac{1039133580104}{1041753338401}$, $\frac{50067778680}{1041753338401}a^{31}+\frac{83981199424}{1041753338401}a^{30}-\frac{360488006496}{1041753338401}a^{29}-\frac{661748403928}{1041753338401}a^{28}+\frac{1885051867302}{1041753338401}a^{27}+\frac{3614695052240}{1041753338401}a^{26}-\frac{8671730633480}{1041753338401}a^{25}-\frac{17026265438577}{1041753338401}a^{24}+\frac{37390617118224}{1041753338401}a^{23}+\frac{74420299706752}{1041753338401}a^{22}-\frac{103019461411968}{1041753338401}a^{21}-\frac{223998050769320}{1041753338401}a^{20}+\frac{255182950987290}{1041753338401}a^{19}+\frac{582970070021264}{1041753338401}a^{18}-\frac{568028962680336}{1041753338401}a^{17}-\frac{13\!\cdots\!61}{1041753338401}a^{16}+\frac{10\!\cdots\!48}{1041753338401}a^{15}+\frac{27\!\cdots\!52}{1041753338401}a^{14}-\frac{10\!\cdots\!68}{1041753338401}a^{13}-\frac{36\!\cdots\!00}{1041753338401}a^{12}+\frac{11\!\cdots\!90}{1041753338401}a^{11}+\frac{45\!\cdots\!84}{1041753338401}a^{10}-\frac{10\!\cdots\!56}{1041753338401}a^{9}-\frac{49\!\cdots\!16}{1041753338401}a^{8}+\frac{68783114350584}{1041753338401}a^{7}+\frac{36\!\cdots\!68}{1041753338401}a^{6}+\frac{16\!\cdots\!85}{1041753338401}a^{5}+\frac{201477975640668}{1041753338401}a^{4}+\frac{317930394618}{1041753338401}a^{3}+\frac{1551113226768}{1041753338401}a^{2}-\frac{20027111472}{1041753338401}a-\frac{99383941280}{1041753338401}$, $\frac{9366808}{1041753338401}a^{31}-\frac{64413963512}{1041753338401}a^{30}+\frac{520573258412}{1041753338401}a^{28}-\frac{2875994092528}{1041753338401}a^{26}+\frac{13627073723695}{1041753338401}a^{24}-\frac{756474321208}{13186751119}a^{22}+\frac{9841870577}{1041753338401}a^{21}+\frac{183539861141312}{1041753338401}a^{20}-\frac{482732498250528}{1041753338401}a^{18}+\frac{11\!\cdots\!88}{1041753338401}a^{16}-\frac{22\!\cdots\!64}{1041753338401}a^{14}+\frac{32\!\cdots\!24}{1041753338401}a^{12}+\frac{985803563850}{1041753338401}a^{11}-\frac{40\!\cdots\!56}{1041753338401}a^{10}+\frac{57226634181183}{13186751119}a^{8}-\frac{35\!\cdots\!72}{1041753338401}a^{6}+\frac{469618395048388}{1041753338401}a^{4}-\frac{16712672970832}{1041753338401}a^{2}+\frac{7912903653600}{1041753338401}a+\frac{1133263304192}{1041753338401}$, $\frac{361033085296}{1041753338401}a^{31}-\frac{6579480000}{1041753338401}a^{30}-\frac{2865700114537}{1041753338401}a^{29}+\frac{47372256000}{1041753338401}a^{28}+\frac{15704939447477}{1041753338401}a^{27}-\frac{247717422000}{1041753338401}a^{26}-\frac{74102040757004}{1041753338401}a^{25}+\frac{1139562526801}{1041753338401}a^{24}+\frac{324207710595808}{1041753338401}a^{23}-\frac{4913555664000}{1041753338401}a^{22}-\frac{981671523487655}{1041753338401}a^{21}+\frac{13537938048000}{1041753338401}a^{20}+\frac{25\!\cdots\!56}{1041753338401}a^{19}-\frac{33533964690000}{1041753338401}a^{18}-\frac{59\!\cdots\!93}{1041753338401}a^{17}+\frac{74645516496000}{1041753338401}a^{16}+\frac{11\!\cdots\!28}{1041753338401}a^{15}-\frac{138181955344920}{1041753338401}a^{14}-\frac{16\!\cdots\!25}{1041753338401}a^{13}+\frac{135521497248000}{1041753338401}a^{12}+\frac{20\!\cdots\!36}{1041753338401}a^{11}-\frac{155797151790000}{1041753338401}a^{10}-\frac{22\!\cdots\!64}{1041753338401}a^{9}+\frac{132637053216000}{1041753338401}a^{8}+\frac{16\!\cdots\!16}{1041753338401}a^{7}-\frac{9038889624000}{1041753338401}a^{6}-\frac{103639060047783}{1041753338401}a^{5}-\frac{223697106722820}{1041753338401}a^{4}+\frac{7040145163272}{1041753338401}a^{3}-\frac{41779698000}{1041753338401}a^{2}-\frac{451291356620}{1041753338401}a+\frac{1044385130401}{1041753338401}$, $\frac{6579480000}{1041753338401}a^{31}+\frac{83981199424}{1041753338401}a^{30}-\frac{47372256000}{1041753338401}a^{29}-\frac{661748403928}{1041753338401}a^{28}+\frac{247717422000}{1041753338401}a^{27}+\frac{3614695052240}{1041753338401}a^{26}-\frac{1139562526801}{1041753338401}a^{25}-\frac{17026265438577}{1041753338401}a^{24}+\frac{4913555664000}{1041753338401}a^{23}+\frac{74420299706752}{1041753338401}a^{22}-\frac{13537938048000}{1041753338401}a^{21}-\frac{223998050769320}{1041753338401}a^{20}+\frac{33533964690000}{1041753338401}a^{19}+\frac{582970070021264}{1041753338401}a^{18}-\frac{74645516496000}{1041753338401}a^{17}-\frac{13\!\cdots\!61}{1041753338401}a^{16}+\frac{138181955344920}{1041753338401}a^{15}+\frac{27\!\cdots\!52}{1041753338401}a^{14}-\frac{135521497248000}{1041753338401}a^{13}-\frac{36\!\cdots\!00}{1041753338401}a^{12}+\frac{155797151790000}{1041753338401}a^{11}+\frac{45\!\cdots\!84}{1041753338401}a^{10}-\frac{132637053216000}{1041753338401}a^{9}-\frac{49\!\cdots\!16}{1041753338401}a^{8}+\frac{9038889624000}{1041753338401}a^{7}+\frac{36\!\cdots\!68}{1041753338401}a^{6}+\frac{223697106722820}{1041753338401}a^{5}+\frac{201477975640668}{1041753338401}a^{4}+\frac{41779698000}{1041753338401}a^{3}+\frac{1551113226768}{1041753338401}a^{2}-\frac{2631792000}{1041753338401}a-\frac{99383941280}{1041753338401}$, $\frac{7333124}{1041753338401}a^{31}+\frac{171204088}{1041753338401}a^{30}-\frac{941622484}{1041753338401}a^{28}+\frac{4451306288}{1041753338401}a^{26}-\frac{19495865521}{1041753338401}a^{24}+\frac{81838660680}{1041753338401}a^{22}+\frac{7700149313}{1041753338401}a^{21}-\frac{155624515992}{1041753338401}a^{20}+\frac{364322299264}{1041753338401}a^{18}-\frac{733117305327}{1041753338401}a^{16}+\frac{1009076894672}{1041753338401}a^{14}+\frac{919609660376}{1041753338401}a^{12}+\frac{764747994683}{1041753338401}a^{11}+\frac{1402846297072}{1041753338401}a^{10}-\frac{1087809374641}{1041753338401}a^{8}+\frac{74131370104}{1041753338401}a^{6}-\frac{5050520596}{1041753338401}a^{4}+\frac{15286898838064}{1041753338401}a^{2}+\frac{4858637081909}{1041753338401}a+\frac{1041731937890}{1041753338401}$, $\frac{64413963512}{1041753338401}a^{30}-\frac{520573258412}{1041753338401}a^{28}+\frac{2875994092528}{1041753338401}a^{26}-\frac{13627073723695}{1041753338401}a^{24}+\frac{756474321208}{13186751119}a^{22}-\frac{183539861141312}{1041753338401}a^{20}+\frac{482732498250528}{1041753338401}a^{18}-\frac{11\!\cdots\!88}{1041753338401}a^{16}+\frac{22\!\cdots\!64}{1041753338401}a^{14}-\frac{32\!\cdots\!24}{1041753338401}a^{12}+\frac{40\!\cdots\!56}{1041753338401}a^{10}-\frac{57226634181183}{13186751119}a^{8}+\frac{35\!\cdots\!72}{1041753338401}a^{6}-\frac{469618395048388}{1041753338401}a^{4}+\frac{16712672970832}{1041753338401}a^{2}+a-\frac{1133263304192}{1041753338401}$, $\frac{244435820}{13186751119}a^{31}-\frac{71164647600}{1041753338401}a^{30}-\frac{1759937904}{13186751119}a^{29}+\frac{568887136896}{1041753338401}a^{28}+\frac{9203008623}{13186751119}a^{27}-\frac{3128162820816}{1041753338401}a^{26}-\frac{42336193885}{13186751119}a^{25}+\frac{14786132116017}{1041753338401}a^{24}+\frac{182544670376}{13186751119}a^{23}-\frac{64756865700112}{1041753338401}a^{22}-\frac{502951143232}{13186751119}a^{21}+\frac{197233423705304}{1041753338401}a^{20}+\frac{1245828265585}{13186751119}a^{19}-\frac{516630785239792}{1041753338401}a^{18}-\frac{2773173265064}{13186751119}a^{17}+\frac{12\!\cdots\!15}{1041753338401}a^{16}+\frac{5133590395152}{13186751119}a^{15}-\frac{24\!\cdots\!56}{1041753338401}a^{14}-\frac{5034791246032}{13186751119}a^{13}+\frac{33\!\cdots\!48}{1041753338401}a^{12}+\frac{5788056890735}{13186751119}a^{11}-\frac{42\!\cdots\!28}{1041753338401}a^{10}-\frac{4927630582544}{13186751119}a^{9}+\frac{46\!\cdots\!98}{1041753338401}a^{8}+\frac{335805929516}{13186751119}a^{7}-\frac{36\!\cdots\!76}{1041753338401}a^{6}+\frac{8301833234116}{13186751119}a^{5}+\frac{245926338846164}{1041753338401}a^{4}+\frac{1552167457}{13186751119}a^{3}-\frac{32041351506896}{1041753338401}a^{2}-\frac{97774328}{13186751119}a+\frac{1135916496703}{1041753338401}$, $\frac{2033684}{1041753338401}a^{31}-\frac{171204088}{1041753338401}a^{30}+\frac{941622484}{1041753338401}a^{28}-\frac{4451306288}{1041753338401}a^{26}+\frac{19495865521}{1041753338401}a^{24}-\frac{81838660680}{1041753338401}a^{22}+\frac{2141721264}{1041753338401}a^{21}+\frac{155624515992}{1041753338401}a^{20}-\frac{364322299264}{1041753338401}a^{18}+\frac{733117305327}{1041753338401}a^{16}-\frac{1009076894672}{1041753338401}a^{14}-\frac{919609660376}{1041753338401}a^{12}+\frac{221055569167}{1041753338401}a^{11}-\frac{1402846297072}{1041753338401}a^{10}+\frac{1087809374641}{1041753338401}a^{8}-\frac{74131370104}{1041753338401}a^{6}+\frac{5050520596}{1041753338401}a^{4}-\frac{15286898838064}{1041753338401}a^{2}+\frac{3054266571691}{1041753338401}a-\frac{1041731937890}{1041753338401}$
| 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: | \( 1749050137699.8193 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 1749050137699.8193 \cdot 80}{10\cdot\sqrt{203282392447840896882957090816000000000000000000000000}}\cr\approx \mathstrut & 0.183113455442011 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^30 + 44*x^28 - 208*x^26 + 911*x^24 - 2776*x^22 + 7272*x^20 - 17024*x^18 + 34257*x^16 - 47152*x^14 + 59688*x^12 - 65552*x^10 + 50831*x^8 - 3464*x^6 + 236*x^4 - 16*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^32 - 8*x^30 + 44*x^28 - 208*x^26 + 911*x^24 - 2776*x^22 + 7272*x^20 - 17024*x^18 + 34257*x^16 - 47152*x^14 + 59688*x^12 - 65552*x^10 + 50831*x^8 - 3464*x^6 + 236*x^4 - 16*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^32 - 8*x^30 + 44*x^28 - 208*x^26 + 911*x^24 - 2776*x^22 + 7272*x^20 - 17024*x^18 + 34257*x^16 - 47152*x^14 + 59688*x^12 - 65552*x^10 + 50831*x^8 - 3464*x^6 + 236*x^4 - 16*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^32 - 8*x^30 + 44*x^28 - 208*x^26 + 911*x^24 - 2776*x^22 + 7272*x^20 - 17024*x^18 + 34257*x^16 - 47152*x^14 + 59688*x^12 - 65552*x^10 + 50831*x^8 - 3464*x^6 + 236*x^4 - 16*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
$C_2\times C_4^2$ (as 32T36):
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 32 |
| The 32 conjugacy class representatives for $C_2\times C_4^2$ |
| Character table for $C_2\times C_4^2$ |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{8}$ | ${\href{/padicField/11.4.0.1}{4} }^{8}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.4.0.1}{4} }^{8}$ | ${\href{/padicField/19.4.0.1}{4} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{8}$ | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.2.0.1}{2} }^{16}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $32$ | $8$ | $4$ | $96$ | |||
|
\(3\)
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(5\)
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |
| 5.16.12.1 | $x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$ | $4$ | $4$ | $12$ | $C_4^2$ | $[\ ]_{4}^{4}$ |