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Properties

Label 36.0.50263695010...8933.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{18}\cdot 7^{30}\cdot 13^{33}$
Root discriminant $92.03$
Ramified primes $3, 7, 13$
Class number Not computed
Class group Not computed
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![21305513503, -12457929369, 45711370125, -8083073282, 45670623647, -1674794190, 30454580477, -1760009807, 17005484365, -3127376355, 8327023718, -2754343527, 3605252587, -1601650713, 1351689873, -703532376, 444193428, -246519877, 138221149, -76386185, 45074515, -22388250, 15017355, -5825079, 4408145, -1208558, 1002183, -185158, 163941, -19781, 18345, -1379, 1322, -56, 55, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 55*x^34 - 56*x^33 + 1322*x^32 - 1379*x^31 + 18345*x^30 - 19781*x^29 + 163941*x^28 - 185158*x^27 + 1002183*x^26 - 1208558*x^25 + 4408145*x^24 - 5825079*x^23 + 15017355*x^22 - 22388250*x^21 + 45074515*x^20 - 76386185*x^19 + 138221149*x^18 - 246519877*x^17 + 444193428*x^16 - 703532376*x^15 + 1351689873*x^14 - 1601650713*x^13 + 3605252587*x^12 - 2754343527*x^11 + 8327023718*x^10 - 3127376355*x^9 + 17005484365*x^8 - 1760009807*x^7 + 30454580477*x^6 - 1674794190*x^5 + 45670623647*x^4 - 8083073282*x^3 + 45711370125*x^2 - 12457929369*x + 21305513503)
 
gp: K = bnfinit(x^36 - x^35 + 55*x^34 - 56*x^33 + 1322*x^32 - 1379*x^31 + 18345*x^30 - 19781*x^29 + 163941*x^28 - 185158*x^27 + 1002183*x^26 - 1208558*x^25 + 4408145*x^24 - 5825079*x^23 + 15017355*x^22 - 22388250*x^21 + 45074515*x^20 - 76386185*x^19 + 138221149*x^18 - 246519877*x^17 + 444193428*x^16 - 703532376*x^15 + 1351689873*x^14 - 1601650713*x^13 + 3605252587*x^12 - 2754343527*x^11 + 8327023718*x^10 - 3127376355*x^9 + 17005484365*x^8 - 1760009807*x^7 + 30454580477*x^6 - 1674794190*x^5 + 45670623647*x^4 - 8083073282*x^3 + 45711370125*x^2 - 12457929369*x + 21305513503, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 55 x^{34} - 56 x^{33} + 1322 x^{32} - 1379 x^{31} + 18345 x^{30} - 19781 x^{29} + 163941 x^{28} - 185158 x^{27} + 1002183 x^{26} - 1208558 x^{25} + 4408145 x^{24} - 5825079 x^{23} + 15017355 x^{22} - 22388250 x^{21} + 45074515 x^{20} - 76386185 x^{19} + 138221149 x^{18} - 246519877 x^{17} + 444193428 x^{16} - 703532376 x^{15} + 1351689873 x^{14} - 1601650713 x^{13} + 3605252587 x^{12} - 2754343527 x^{11} + 8327023718 x^{10} - 3127376355 x^{9} + 17005484365 x^{8} - 1760009807 x^{7} + 30454580477 x^{6} - 1674794190 x^{5} + 45670623647 x^{4} - 8083073282 x^{3} + 45711370125 x^{2} - 12457929369 x + 21305513503 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(50263695010347434765870190422659414897457208738397988679495618135108933=3^{18}\cdot 7^{30}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.03$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(4,·)$, $\chi_{273}(5,·)$, $\chi_{273}(142,·)$, $\chi_{273}(16,·)$, $\chi_{273}(20,·)$, $\chi_{273}(22,·)$, $\chi_{273}(25,·)$, $\chi_{273}(164,·)$, $\chi_{273}(167,·)$, $\chi_{273}(41,·)$, $\chi_{273}(43,·)$, $\chi_{273}(172,·)$, $\chi_{273}(47,·)$, $\chi_{273}(59,·)$, $\chi_{273}(188,·)$, $\chi_{273}(64,·)$, $\chi_{273}(205,·)$, $\chi_{273}(206,·)$, $\chi_{273}(79,·)$, $\chi_{273}(80,·)$, $\chi_{273}(83,·)$, $\chi_{273}(215,·)$, $\chi_{273}(88,·)$, $\chi_{273}(89,·)$, $\chi_{273}(227,·)$, $\chi_{273}(100,·)$, $\chi_{273}(235,·)$, $\chi_{273}(236,·)$, $\chi_{273}(110,·)$, $\chi_{273}(211,·)$, $\chi_{273}(121,·)$, $\chi_{273}(122,·)$, $\chi_{273}(125,·)$, $\chi_{273}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{1651774123} a^{31} + \frac{804093161}{1651774123} a^{30} - \frac{215371445}{1651774123} a^{29} + \frac{739070381}{1651774123} a^{28} - \frac{239098284}{1651774123} a^{27} - \frac{228601195}{1651774123} a^{26} + \frac{256389141}{1651774123} a^{25} + \frac{728776042}{1651774123} a^{24} + \frac{821908761}{1651774123} a^{23} + \frac{388645305}{1651774123} a^{22} - \frac{170312026}{1651774123} a^{21} - \frac{60013352}{1651774123} a^{20} + \frac{681964989}{1651774123} a^{19} - \frac{436485775}{1651774123} a^{18} + \frac{522086939}{1651774123} a^{17} + \frac{193420413}{1651774123} a^{16} - \frac{327803534}{1651774123} a^{15} + \frac{3124728}{1651774123} a^{14} + \frac{671817537}{1651774123} a^{13} - \frac{487708371}{1651774123} a^{12} - \frac{486654167}{1651774123} a^{11} - \frac{117199552}{1651774123} a^{10} + \frac{56256849}{1651774123} a^{9} + \frac{145159999}{1651774123} a^{8} + \frac{344505488}{1651774123} a^{7} + \frac{682027502}{1651774123} a^{6} + \frac{430413957}{1651774123} a^{5} + \frac{243296748}{1651774123} a^{4} + \frac{755432998}{1651774123} a^{3} - \frac{177012705}{1651774123} a^{2} + \frac{399939905}{1651774123} a + \frac{86935166}{1651774123}$, $\frac{1}{4955322369} a^{32} - \frac{471153734}{4955322369} a^{30} + \frac{493332882}{1651774123} a^{29} - \frac{526185065}{4955322369} a^{28} + \frac{140497541}{1651774123} a^{27} + \frac{225991919}{4955322369} a^{26} + \frac{2024939917}{4955322369} a^{25} - \frac{72971251}{1651774123} a^{24} - \frac{1085838056}{4955322369} a^{23} - \frac{575770684}{1651774123} a^{22} - \frac{220391638}{1651774123} a^{21} - \frac{562717840}{1651774123} a^{20} + \frac{2382978020}{4955322369} a^{19} - \frac{2109142702}{4955322369} a^{18} - \frac{297027376}{4955322369} a^{17} + \frac{359684129}{4955322369} a^{16} + \frac{130300625}{1651774123} a^{15} + \frac{1339396057}{4955322369} a^{14} + \frac{284848333}{1651774123} a^{13} + \frac{1655473552}{4955322369} a^{12} - \frac{587814184}{4955322369} a^{11} + \frac{2114166553}{4955322369} a^{10} - \frac{282360256}{4955322369} a^{9} + \frac{2061749824}{4955322369} a^{8} - \frac{148534526}{1651774123} a^{7} + \frac{153555683}{1651774123} a^{6} - \frac{446052067}{4955322369} a^{5} - \frac{667309696}{4955322369} a^{4} + \frac{263214156}{1651774123} a^{3} + \frac{1695245518}{4955322369} a^{2} - \frac{2068383512}{4955322369} a + \frac{412386694}{4955322369}$, $\frac{1}{4955322369} a^{33} + \frac{1}{4955322369} a^{31} + \frac{548519854}{1651774123} a^{30} - \frac{1460071430}{4955322369} a^{29} + \frac{160069137}{1651774123} a^{28} - \frac{292208347}{4955322369} a^{27} + \frac{464856343}{4955322369} a^{26} - \frac{626739447}{1651774123} a^{25} - \frac{2270098292}{4955322369} a^{24} - \frac{20175011}{1651774123} a^{23} - \frac{736569857}{1651774123} a^{22} + \frac{567612888}{1651774123} a^{21} - \frac{2100486859}{4955322369} a^{20} - \frac{2445144634}{4955322369} a^{19} + \frac{522853703}{4955322369} a^{18} - \frac{1618671982}{4955322369} a^{17} - \frac{521410107}{1651774123} a^{16} - \frac{754970834}{4955322369} a^{15} + \frac{615591393}{1651774123} a^{14} + \frac{402964087}{4955322369} a^{13} + \frac{1041761513}{4955322369} a^{12} - \frac{841862525}{4955322369} a^{11} - \frac{1454476339}{4955322369} a^{10} - \frac{1293883010}{4955322369} a^{9} - \frac{82202797}{1651774123} a^{8} - \frac{35688734}{1651774123} a^{7} - \frac{1984779280}{4955322369} a^{6} + \frac{896038754}{4955322369} a^{5} - \frac{581473358}{1651774123} a^{4} - \frac{2282200115}{4955322369} a^{3} + \frac{2131979614}{4955322369} a^{2} + \frac{879048151}{4955322369} a + \frac{214335195}{1651774123}$, $\frac{1}{7823194838549780517} a^{34} - \frac{248999600}{7823194838549780517} a^{33} + \frac{156561776}{7823194838549780517} a^{32} + \frac{2122915372}{7823194838549780517} a^{31} + \frac{3884003325679496828}{7823194838549780517} a^{30} + \frac{3116799955815850801}{7823194838549780517} a^{29} + \frac{49694056594082633}{869243870949975613} a^{28} - \frac{1238879077507285132}{2607731612849926839} a^{27} + \frac{196926773722466545}{869243870949975613} a^{26} + \frac{1301363339254265504}{7823194838549780517} a^{25} + \frac{927216439615430590}{7823194838549780517} a^{24} + \frac{3423764573319384625}{7823194838549780517} a^{23} - \frac{749006950430019826}{2607731612849926839} a^{22} + \frac{1876965572709462227}{7823194838549780517} a^{21} - \frac{2763353443851217442}{7823194838549780517} a^{20} + \frac{254728986729018224}{2607731612849926839} a^{19} + \frac{574196795104371941}{2607731612849926839} a^{18} - \frac{1229456898221341943}{7823194838549780517} a^{17} - \frac{242179601636600124}{869243870949975613} a^{16} - \frac{1699240079330349818}{7823194838549780517} a^{15} - \frac{1992850140209288374}{7823194838549780517} a^{14} + \frac{446045615717723515}{2607731612849926839} a^{13} + \frac{1215189293779622302}{7823194838549780517} a^{12} - \frac{3211011959212684921}{7823194838549780517} a^{11} - \frac{2296096243868524895}{7823194838549780517} a^{10} - \frac{413527648369360963}{869243870949975613} a^{9} + \frac{572004952536902176}{7823194838549780517} a^{8} - \frac{161856576206430469}{7823194838549780517} a^{7} - \frac{3785178679005114746}{7823194838549780517} a^{6} - \frac{2243754670511213957}{7823194838549780517} a^{5} - \frac{42711139235757365}{869243870949975613} a^{4} + \frac{2706892185825564044}{7823194838549780517} a^{3} - \frac{289487449470868185}{869243870949975613} a^{2} - \frac{61549059152815207}{7823194838549780517} a + \frac{342702980572323022}{7823194838549780517}$, $\frac{1}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{35} - \frac{281844677376578486794227026717749629250486993830896499292562429738551780604853125296312331025653700496524297754364997419800}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{34} - \frac{196653091114777503660718651092229326267401124785996822368299088964689819036654070543715078739642397362381722389825517797762330075003}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{33} - \frac{93596400558079233093391175205829380559809016902337697182309985559650157482578516410186721938772264733945352313373989720115349000349}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{32} - \frac{174553047530742006553543325825949615896851132309050335660429679723248707942178340031111759671986843912318942544043021336165190329985}{976768287298118757349881637240952274337952876638930565787022950150532658696967196582584158034148802256691506407819888236772537832637545805771} a^{31} - \frac{536507318835056364868311774980948372894872783921630378213200020835494841350482200860134164806858347397912087054006292346513685259855694731670}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{30} - \frac{3392753079417192874033909185621289919525043939709668226194222590400713605993550682809542994476271482110028749513904237993312630951009925101721}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{29} + \frac{1147043449403519093506464297459230548303976332252719724561194580141269323506778555153783789742083083842309576701819872808902469158117456526210}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{28} - \frac{274586182712399044431710234127843143117859195609636133879426533358742507814160398968269537748684026559574247127249940483832285791873991606747}{976768287298118757349881637240952274337952876638930565787022950150532658696967196582584158034148802256691506407819888236772537832637545805771} a^{27} + \frac{1017183557907767968520185188234474112911460240581790148327033319800556978078541249019840737594169183293210792345931831214418695450889731743111}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{26} - \frac{50784950232755472051412279938401191562984518953378756041657880111250081461407442567052165614601377225450085667180076802087073418335474816308}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{25} + \frac{1409115027991250044715188716650307795033001184399158776782396319757522625440394366482604941104922115281070822497789989669844374486870126810521}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{24} - \frac{1869686112413740519001127572509885201915368051485495499745005026490479143584110882187865111508340397632053876564800536588478498750817269309019}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{23} - \frac{3106742815042828339900729785539714853285598393624759757964553255945461860083445203910132014706362116954154628293110191579961707650865412218422}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{22} + \frac{10387387639042703980498578792977344344734311509188219046098956251830416220693852458147983679757490178154728305773712952258697000119412231398}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{21} - \frac{2137239042832516941176698116798864371175693280254561445137169440420363535612891776211646552153077085600474836386561755517211578105137966741078}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{20} - \frac{1386113600289698880622709923718400701366243663931913464707280268144136843493964982924074318935853294282039555781908354136402419112841312919075}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{19} - \frac{5973382952321857357699772846743124649568443730718405055395531922642267303702709284889031877178349328958423392162726208649229851260976456246}{16071141838543087415263134799211280564975458664991544958104582360794870069968381662236302417380876088318507418044568545029164242218899291137} a^{18} - \frac{4128700051477186303118253546251693039567403511913252847245394221781831337155966802459824601768523319748671713912019250717549202704494042488171}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{17} + \frac{1889512596497420113587808698965923733913524831219821648565357320314946437509107526301219500713510227900971937905225332706555743388156750021311}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{16} + \frac{320283711306962330050364733087691829530572729733636994970400922664842448297406318323506151543637951477865431923196151402984585119770177725272}{976768287298118757349881637240952274337952876638930565787022950150532658696967196582584158034148802256691506407819888236772537832637545805771} a^{15} - \frac{64068257752470594042128288535076180070900173473261653942114618152746949990057798748155078017999600758696862691834057347248547687145358688544}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{14} + \frac{366933435000067067959717167601243684967578308048329479557060422166288712275976585425210224394995367569307402081367315162458276222864202020004}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{13} + \frac{1276037491063868164496927076948317756028796323623662233630298742591975264821125352128221033298926840377454214508659745872785999812758870523436}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{12} + \frac{1356357573312195651569171450511908587408763525037343122641847974312420555932816929418349451703198122430248244962378666020602679200874563446928}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{11} + \frac{3275119884189272366595553061331018142955425051743435993298436541552860156566312580883316132082191700229041707272407379893939405458625158314313}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{10} + \frac{1838116153320086966409837132300506877665475529758655031344389255936840192465139412508945414453656649548372415177874205169467757542499473459191}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{9} + \frac{1110943037593524224052707302367963833417341840881717424086484234192187380942469106919067959716412725839257149775197599203342843194548776586792}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{8} - \frac{1133533710929666078068718618003917114226409078525838007564705380587268493435653953192983143591079105882727501898536558615781600021290908875202}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a^{7} + \frac{2147945918188859771925492654597280030703462852692950168785447257289364316700878470878413139517967275597946543391421473888484711857400246519899}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{6} - \frac{2523828308432975828954806019867155776542322080094036282096298537743116639675964129769904503050536633035922114051070444450949387039193716385297}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{5} + \frac{1553236186160600619609988367108285634465047174108191643257515706929961611217600763726806861040667725725125793926144198961016813424038856451154}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{4} - \frac{589069657082228139528143521320542198742224013258512117531244035193508935543605398870944230930219684894341877998167529692374838226415007806206}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{3} + \frac{1062900963539807879886481636605707287498757386267320056657817625159744230861610586459133376813259415479850240574885362502780787415554176427081}{8790914585683068816148934735168570469041575889750375092083206551354793928272704769243257422307339220310223557670378994130952840493737912251939} a^{2} + \frac{1339680792977368780745970809045790874141588075266443784579747294879048918489825414100430846781580513301463685697584308690923900919727120317902}{2930304861894356272049644911722856823013858629916791697361068850451597976090901589747752474102446406770074519223459664710317613497912637417313} a - \frac{180426583418938198308629927713418390770998339100334819947949105674838730965701192339111607529239275766011711804557326650807633484234}{412612190006367705999192726200708201209956816394991120157617097136942633974153069219420513598467254943882098539363188712295682258813}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.8281.1, 4.0.968877.2, \(\Q(\zeta_{13})^+\), 6.6.891474493.2, 6.6.5274997.1, 6.6.891474493.1, 9.9.567869252041.1, 12.0.153706645206385020477.1, 12.0.369049655140530434165277.2, 12.0.2183725770062310261333.1, 12.0.369049655140530434165277.1, 18.18.708478645847689707516501157.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/11.12.0.1}{12} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$