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Properties

Label 42.42.5389501803...8413.1
Degree $42$
Signature $[42, 0]$
Discriminant $13^{21}\cdot 43^{40}$
Root discriminant $129.62$
Ramified primes $13, 43$
Class number Not computed
Class group Not computed
Galois group $C_{42}$ (as 42T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44461, 1039861, -30584885, 244358694, -610194213, -2197742173, 18233137259, -39826718532, -20445279017, 253863699706, -383301896284, -208132634006, 1219560360804, -902235266041, -1169869610784, 2136571761785, -118443932597, -1945232843680, 1066146428349, 846517600196, -932184067908, -103722614436, 422057640901, -73238296287, -114224542965, 43891669705, 18144473160, -12031497467, -1223049156, 2001195013, -118867851, -210312450, 37396197, 13158257, -4111051, -354937, 242768, -9004, -7330, 912, 67, -19, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 19*x^41 + 67*x^40 + 912*x^39 - 7330*x^38 - 9004*x^37 + 242768*x^36 - 354937*x^35 - 4111051*x^34 + 13158257*x^33 + 37396197*x^32 - 210312450*x^31 - 118867851*x^30 + 2001195013*x^29 - 1223049156*x^28 - 12031497467*x^27 + 18144473160*x^26 + 43891669705*x^25 - 114224542965*x^24 - 73238296287*x^23 + 422057640901*x^22 - 103722614436*x^21 - 932184067908*x^20 + 846517600196*x^19 + 1066146428349*x^18 - 1945232843680*x^17 - 118443932597*x^16 + 2136571761785*x^15 - 1169869610784*x^14 - 902235266041*x^13 + 1219560360804*x^12 - 208132634006*x^11 - 383301896284*x^10 + 253863699706*x^9 - 20445279017*x^8 - 39826718532*x^7 + 18233137259*x^6 - 2197742173*x^5 - 610194213*x^4 + 244358694*x^3 - 30584885*x^2 + 1039861*x + 44461)
 
gp: K = bnfinit(x^42 - 19*x^41 + 67*x^40 + 912*x^39 - 7330*x^38 - 9004*x^37 + 242768*x^36 - 354937*x^35 - 4111051*x^34 + 13158257*x^33 + 37396197*x^32 - 210312450*x^31 - 118867851*x^30 + 2001195013*x^29 - 1223049156*x^28 - 12031497467*x^27 + 18144473160*x^26 + 43891669705*x^25 - 114224542965*x^24 - 73238296287*x^23 + 422057640901*x^22 - 103722614436*x^21 - 932184067908*x^20 + 846517600196*x^19 + 1066146428349*x^18 - 1945232843680*x^17 - 118443932597*x^16 + 2136571761785*x^15 - 1169869610784*x^14 - 902235266041*x^13 + 1219560360804*x^12 - 208132634006*x^11 - 383301896284*x^10 + 253863699706*x^9 - 20445279017*x^8 - 39826718532*x^7 + 18233137259*x^6 - 2197742173*x^5 - 610194213*x^4 + 244358694*x^3 - 30584885*x^2 + 1039861*x + 44461, 1)
 

Normalized defining polynomial

\( x^{42} - 19 x^{41} + 67 x^{40} + 912 x^{39} - 7330 x^{38} - 9004 x^{37} + 242768 x^{36} - 354937 x^{35} - 4111051 x^{34} + 13158257 x^{33} + 37396197 x^{32} - 210312450 x^{31} - 118867851 x^{30} + 2001195013 x^{29} - 1223049156 x^{28} - 12031497467 x^{27} + 18144473160 x^{26} + 43891669705 x^{25} - 114224542965 x^{24} - 73238296287 x^{23} + 422057640901 x^{22} - 103722614436 x^{21} - 932184067908 x^{20} + 846517600196 x^{19} + 1066146428349 x^{18} - 1945232843680 x^{17} - 118443932597 x^{16} + 2136571761785 x^{15} - 1169869610784 x^{14} - 902235266041 x^{13} + 1219560360804 x^{12} - 208132634006 x^{11} - 383301896284 x^{10} + 253863699706 x^{9} - 20445279017 x^{8} - 39826718532 x^{7} + 18233137259 x^{6} - 2197742173 x^{5} - 610194213 x^{4} + 244358694 x^{3} - 30584885 x^{2} + 1039861 x + 44461 \)

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

Invariants

Degree:  $42$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[42, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(53895018039694418583693741474726125588548507344558968174790387731623712959431675466748413=13^{21}\cdot 43^{40}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.62$
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:  $13, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(559=13\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{559}(1,·)$, $\chi_{559}(259,·)$, $\chi_{559}(391,·)$, $\chi_{559}(324,·)$, $\chi_{559}(14,·)$, $\chi_{559}(365,·)$, $\chi_{559}(272,·)$, $\chi_{559}(274,·)$, $\chi_{559}(532,·)$, $\chi_{559}(25,·)$, $\chi_{559}(547,·)$, $\chi_{559}(326,·)$, $\chi_{559}(38,·)$, $\chi_{559}(40,·)$, $\chi_{559}(298,·)$, $\chi_{559}(428,·)$, $\chi_{559}(53,·)$, $\chi_{559}(311,·)$, $\chi_{559}(441,·)$, $\chi_{559}(443,·)$, $\chi_{559}(181,·)$, $\chi_{559}(64,·)$, $\chi_{559}(66,·)$, $\chi_{559}(196,·)$, $\chi_{559}(454,·)$, $\chi_{559}(183,·)$, $\chi_{559}(79,·)$, $\chi_{559}(337,·)$, $\chi_{559}(339,·)$, $\chi_{559}(142,·)$, $\chi_{559}(90,·)$, $\chi_{559}(207,·)$, $\chi_{559}(92,·)$, $\chi_{559}(350,·)$, $\chi_{559}(144,·)$, $\chi_{559}(482,·)$, $\chi_{559}(103,·)$, $\chi_{559}(402,·)$, $\chi_{559}(246,·)$, $\chi_{559}(404,·)$, $\chi_{559}(508,·)$, $\chi_{559}(170,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $\frac{1}{59775013457} a^{40} + \frac{3845516951}{59775013457} a^{39} + \frac{24582110822}{59775013457} a^{38} + \frac{17477751692}{59775013457} a^{37} + \frac{18621141935}{59775013457} a^{36} + \frac{2021997463}{59775013457} a^{35} - \frac{28575949981}{59775013457} a^{34} + \frac{23196676798}{59775013457} a^{33} - \frac{28860836890}{59775013457} a^{32} - \frac{12329561075}{59775013457} a^{31} + \frac{24830226425}{59775013457} a^{30} - \frac{29150613835}{59775013457} a^{29} - \frac{4255329489}{59775013457} a^{28} - \frac{16784453232}{59775013457} a^{27} - \frac{5490442125}{59775013457} a^{26} - \frac{21394202801}{59775013457} a^{25} + \frac{29167229584}{59775013457} a^{24} + \frac{23471829160}{59775013457} a^{23} - \frac{28403326286}{59775013457} a^{22} + \frac{5081802683}{59775013457} a^{21} + \frac{4191470145}{59775013457} a^{20} + \frac{19546845268}{59775013457} a^{19} - \frac{27526029838}{59775013457} a^{18} + \frac{20978766827}{59775013457} a^{17} + \frac{3479560153}{59775013457} a^{16} - \frac{14070163094}{59775013457} a^{15} - \frac{13775485550}{59775013457} a^{14} - \frac{10672725091}{59775013457} a^{13} - \frac{29328372407}{59775013457} a^{12} + \frac{14176900978}{59775013457} a^{11} - \frac{10857053946}{59775013457} a^{10} - \frac{23220625636}{59775013457} a^{9} + \frac{8143029247}{59775013457} a^{8} + \frac{16100542435}{59775013457} a^{7} + \frac{29446637782}{59775013457} a^{6} + \frac{18019114071}{59775013457} a^{5} + \frac{22322748817}{59775013457} a^{4} - \frac{22006793820}{59775013457} a^{3} + \frac{24907739021}{59775013457} a^{2} + \frac{24054892943}{59775013457} a + \frac{344396}{1344437}$, $\frac{1}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{41} + \frac{746942358670852689185073512897111134006695052801312715680270985203237910599153010941481076731781859256874190433907015782946893613290816061245604323772781909365}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{40} + \frac{16558147667612818417098277829758741263556852385224978620710674611988659800554515883942818690387498510062417885918640496355421397201816309302980742526958333503113843649285}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{39} - \frac{12654955591820480620588333329906488276472960470914360009854512341905776073256688578309249406446110425617880668442263375414430859341327272534479085359684687324659007243975}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{38} + \frac{45853212204047917891220856526403817921071516681087605338873156436894715920900392498281223606237004501806628053428623091718003727164276393526144515911297516579313997167854}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{37} + \frac{30881364545846239451783297967302310702486280669871167640198839927426142879701454003026770803662414654099976292002183766906741261147545886421653553325880923399923931531452}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{36} + \frac{43993732651478407630329748495518004482820443330847097644058533509242762331351057116544041738826405712286957616538482367546407293452332128395815465784177070812194292618137}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{35} - \frac{61336105789389301531635991019163676851593850124218987858936014104175095394404965309983740208996145206830205654140089850592702340261849498319288464600410099544375317192901}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{34} + \frac{51753110343074784375484103918109358331826049812686724816896531554472915239633322609460749934283388141573158130157992435006887013569081579865737491806066726310111431925102}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{33} - \frac{58130446821201219162975358771102267512225511467093118949551607900253649030332464965930324407368234600338718619677657501063198291271894772578662579941241241629065348205379}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{32} + \frac{28651420018162595040374830259861217383600166892403929623817782999426398160353348869161085617500733051801732553283050465920285693788355073332161432391444288260305065576133}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{31} + \frac{36723424881388034814402912462620270246453882646663727935474113227438758607750254590051619158779713945396375699432535303487246021442407955431651151284372907282537642533479}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{30} - \frac{1992174706880579982460639717793229546827891469959558100353711516850826449446366254171066922795162544845083311209895276935139126169883752359393765078333712799585242061828}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{29} + \frac{6050834867299155132883943928399739336706838637624702193223763824532615977039651547911617416197268832981103366189579639127429898584676515939276855210102621385190664427421}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{28} - \frac{4988841726919714564182534090530526535141451846344418058169895143567767493737102732919455628481451125561607209819921357521064834466345470429006742391270306503891259291074}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{27} - \frac{11189387469418457282546240697273049534539317719355478047615538838407338785312273110877528911784528475995565450815370983711981206883633493513194791755783221415586012331304}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{26} + \frac{7003325667028216186146078219570848557480686468719868291559330930195262948130232611180763938896818636356321479763033238226894751336046610319854639831452539507109458123098}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{25} - \frac{18706685319998132111207017024753280305764006042623871883307327515023743215470630078006697754490158213599327895583483694622948532592669826944152288257987294291443049625483}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{24} - \frac{27799956902984001844037546371384878711802979109447612288227365396961747156067179864972638972750895503461082663828885083048147162431852323373707473125462678383766220289747}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{23} + \frac{46047006872384188027246032635215542528516523730528117309879249339361353111144331233813939532044195813939633990496107712777647087390710533116628175042051730889946888453765}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{22} + \frac{27534966591176785552221446494012836516099014097305875184517184358506126034336351943838677833992217588615990302798127347267707612916612903080463193956685917388049915365107}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{21} + \frac{31730881084399748203903822963801331933610044520008184901126871878175800325451411038368552956852558647506570075551097832884080095642003223078149953753860484696847878144130}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{20} + \frac{26361558564727651442707495606645977494389896709857830939246902419729235200231443943017118048527245603284205523842634463340230558588712052492263521242635206231239101700779}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{19} - \frac{27312936124431145103754247091539127144148487846294669829177418587582915303502835288921560661114943714433431422416162089845823904070503180416700236619140635591641631692836}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{18} + \frac{17015589335231554792731993085370045301637733577132365776128397617427542361438678057227425120403974266746049316339726832188634830928354095150689677174488950287614272875714}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{17} - \frac{33022868176656130883600928581987575981920351721704304585096224718298187076876266437387661820219730100886050525830407040840002165883720088058900749445730924042317334189385}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{16} + \frac{5756233717908181699107448748593720779269651790271041599109785309047427238845302471216893006868346434903157681188597051330350602878514430509975215398473637218483162833963}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{15} - \frac{38008525585224198304744927805428991058684971622056013496366725560583102231858648155340490732445294553529523165119291424826963307389057094853784366348774426810958460669896}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{14} + \frac{49285429625849857624765781583110194371964143883090853546839196569375524010971578869900833390987385998133004686449684536300854438384519050252252870377194393730906252637000}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{13} + \frac{55893406279400074177320818780934039011716826296846040823116546101687991427674831165926543284186433287129977132093979223572652394923704336782955732045570000409362903300030}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{12} - \frac{47779768047489891388088984200028026359015071702901224733269999467906903690545161561051589690679290520536230909738446008702529382174049969268630551879933417509275581174925}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{11} + \frac{36924274068473020469413801779734142507565794173427541884425881146912323304854387020159267701043737911750391491611098535459013968114732340311477761571883078154717490679596}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{10} - \frac{4841393393441453915295674787608617174094757547298314337241204629191324147208455252958845557346962484455171653234181144187064560429519040656170013051656164076181063609348}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{9} - \frac{50037048528470095909993984010783938745397128883989189171214613450959693817713995767407729211112637191391740169969378472715526477120452779884226327787536125747584378748380}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{8} + \frac{33624320579873361219534997937665418948675903264852801609771878633374486741026490743445668847202301094087525070551804130655198571571348896559691440928942877697253337709821}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{7} + \frac{4672295675151325148924936549270244216900810804208679488497299981690972743768602297926512789913401838383262248068959756031857253288521986630574572891871351669657864272703}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{6} - \frac{26698983326518311154775354963563728558914697883317911080060564394074879192252864839933598585379567783966485268999995029258149177198939387718840062475141143240215368872630}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{5} - \frac{50290811639241751739055111136148953544314333658375047409762300624402644953353456171806994541799831579475524792241954624294893392779869513518768538623059153907090909184165}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{4} - \frac{23066187478971378856111579874514707708972348022068628460585398715987076922769381691537105354166971028554325954228962723363932451110042976357382381943825455728009711029494}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{3} - \frac{29856007674206729443132953008484251414017314590976342835067308393751534397229227449268819005058016525010298128697764525336760939469113631553649710364235909319328503377752}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a^{2} + \frac{13519033704789654389887038242495125801078718839140141875452364491269756184819280404005225295208572927943298062229292062277145039971545682063210653665641497322549904621154}{125114838051958714393643770330378414637302279235132197850623436172479084749943461029975113543608529512112072055575717654828539083930348181890661924257257632211388057892333} a + \frac{1066001014186545472952515627073216024317763736671665663759146213551013687394994343356816679867142164030923953194041850311624911563833352909341343101316085777094433343}{2814035627897679188359320985366465321007226091071550299152592972998337526145238771731969895944952419246352354998216811471369044419386612579354083899535719669179461953}$

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:  $41$
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

$C_{42}$ (as 42T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.1849.1, 6.6.7511105797.1, 7.7.6321363049.1, 14.14.2507407572395754328761947317.1, \(\Q(\zeta_{43})^+\)

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 ${\href{/LocalNumberField/2.14.0.1}{14} }^{3}$ $21^{2}$ $42$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/11.14.0.1}{14} }^{3}$ R $21^{2}$ $42$ $21^{2}$ $21^{2}$ $42$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{3}$ $21^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{3}$

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
13Data not computed
43Data not computed