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Properties

Label 36.36.7476796715...7117.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{48}\cdot 7^{18}\cdot 13^{33}$
Root discriminant $120.18$
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![53, 21021, 1294671, -34504929, 14312169, 2080472373, 2702218315, -17931881889, -12437043336, 64078960801, 19994463534, -123510493746, -9214998883, 143867150712, -11332618518, -107555904357, 19430031183, 53510458269, -13333973109, -18168287730, 5364342456, 4292785887, -1398775770, -714893316, 246806857, 84226455, -29988882, -6965157, 2507526, 394602, -141455, -14553, 5133, 314, -108, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 3*x^35 - 108*x^34 + 314*x^33 + 5133*x^32 - 14553*x^31 - 141455*x^30 + 394602*x^29 + 2507526*x^28 - 6965157*x^27 - 29988882*x^26 + 84226455*x^25 + 246806857*x^24 - 714893316*x^23 - 1398775770*x^22 + 4292785887*x^21 + 5364342456*x^20 - 18168287730*x^19 - 13333973109*x^18 + 53510458269*x^17 + 19430031183*x^16 - 107555904357*x^15 - 11332618518*x^14 + 143867150712*x^13 - 9214998883*x^12 - 123510493746*x^11 + 19994463534*x^10 + 64078960801*x^9 - 12437043336*x^8 - 17931881889*x^7 + 2702218315*x^6 + 2080472373*x^5 + 14312169*x^4 - 34504929*x^3 + 1294671*x^2 + 21021*x + 53)
 
gp: K = bnfinit(x^36 - 3*x^35 - 108*x^34 + 314*x^33 + 5133*x^32 - 14553*x^31 - 141455*x^30 + 394602*x^29 + 2507526*x^28 - 6965157*x^27 - 29988882*x^26 + 84226455*x^25 + 246806857*x^24 - 714893316*x^23 - 1398775770*x^22 + 4292785887*x^21 + 5364342456*x^20 - 18168287730*x^19 - 13333973109*x^18 + 53510458269*x^17 + 19430031183*x^16 - 107555904357*x^15 - 11332618518*x^14 + 143867150712*x^13 - 9214998883*x^12 - 123510493746*x^11 + 19994463534*x^10 + 64078960801*x^9 - 12437043336*x^8 - 17931881889*x^7 + 2702218315*x^6 + 2080472373*x^5 + 14312169*x^4 - 34504929*x^3 + 1294671*x^2 + 21021*x + 53, 1)
 

Normalized defining polynomial

\( x^{36} - 3 x^{35} - 108 x^{34} + 314 x^{33} + 5133 x^{32} - 14553 x^{31} - 141455 x^{30} + 394602 x^{29} + 2507526 x^{28} - 6965157 x^{27} - 29988882 x^{26} + 84226455 x^{25} + 246806857 x^{24} - 714893316 x^{23} - 1398775770 x^{22} + 4292785887 x^{21} + 5364342456 x^{20} - 18168287730 x^{19} - 13333973109 x^{18} + 53510458269 x^{17} + 19430031183 x^{16} - 107555904357 x^{15} - 11332618518 x^{14} + 143867150712 x^{13} - 9214998883 x^{12} - 123510493746 x^{11} + 19994463534 x^{10} + 64078960801 x^{9} - 12437043336 x^{8} - 17931881889 x^{7} + 2702218315 x^{6} + 2080472373 x^{5} + 14312169 x^{4} - 34504929 x^{3} + 1294671 x^{2} + 21021 x + 53 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(747679671598239349782244492011516713922235706186072448481460365111232397117=3^{48}\cdot 7^{18}\cdot 13^{33}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.18$
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:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(1,·)$, $\chi_{819}(643,·)$, $\chi_{819}(769,·)$, $\chi_{819}(265,·)$, $\chi_{819}(400,·)$, $\chi_{819}(43,·)$, $\chi_{819}(274,·)$, $\chi_{819}(22,·)$, $\chi_{819}(538,·)$, $\chi_{819}(673,·)$, $\chi_{819}(34,·)$, $\chi_{819}(547,·)$, $\chi_{819}(295,·)$, $\chi_{819}(811,·)$, $\chi_{819}(307,·)$, $\chi_{819}(568,·)$, $\chi_{819}(316,·)$, $\chi_{819}(64,·)$, $\chi_{819}(580,·)$, $\chi_{819}(202,·)$, $\chi_{819}(76,·)$, $\chi_{819}(589,·)$, $\chi_{819}(337,·)$, $\chi_{819}(211,·)$, $\chi_{819}(475,·)$, $\chi_{819}(349,·)$, $\chi_{819}(223,·)$, $\chi_{819}(97,·)$, $\chi_{819}(610,·)$, $\chi_{819}(484,·)$, $\chi_{819}(748,·)$, $\chi_{819}(622,·)$, $\chi_{819}(496,·)$, $\chi_{819}(370,·)$, $\chi_{819}(757,·)$, $\chi_{819}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $\frac{1}{53} a^{29} - \frac{10}{53} a^{28} - \frac{5}{53} a^{27} + \frac{20}{53} a^{26} - \frac{11}{53} a^{25} - \frac{15}{53} a^{24} + \frac{9}{53} a^{23} - \frac{3}{53} a^{22} + \frac{23}{53} a^{21} - \frac{19}{53} a^{20} - \frac{2}{53} a^{19} + \frac{4}{53} a^{18} - \frac{18}{53} a^{17} - \frac{19}{53} a^{16} + \frac{21}{53} a^{15} - \frac{4}{53} a^{14} + \frac{14}{53} a^{13} - \frac{2}{53} a^{12} + \frac{11}{53} a^{11} - \frac{26}{53} a^{10} - \frac{7}{53} a^{9} + \frac{4}{53} a^{8} + \frac{22}{53} a^{7} + \frac{22}{53} a^{6} + \frac{15}{53} a^{5} + \frac{9}{53} a^{4} + \frac{20}{53} a^{3} - \frac{19}{53} a^{2} - \frac{14}{53} a$, $\frac{1}{53} a^{30} + \frac{1}{53} a^{28} + \frac{23}{53} a^{27} - \frac{23}{53} a^{26} - \frac{19}{53} a^{25} + \frac{18}{53} a^{24} - \frac{19}{53} a^{23} - \frac{7}{53} a^{22} - \frac{1}{53} a^{21} + \frac{20}{53} a^{20} - \frac{16}{53} a^{19} + \frac{22}{53} a^{18} + \frac{13}{53} a^{17} - \frac{10}{53} a^{16} - \frac{6}{53} a^{15} - \frac{26}{53} a^{14} - \frac{21}{53} a^{13} - \frac{9}{53} a^{12} - \frac{22}{53} a^{11} - \frac{2}{53} a^{10} - \frac{13}{53} a^{9} + \frac{9}{53} a^{8} - \frac{23}{53} a^{7} + \frac{23}{53} a^{6} + \frac{4}{53} a^{4} + \frac{22}{53} a^{3} + \frac{8}{53} a^{2} + \frac{19}{53} a$, $\frac{1}{53} a^{31} - \frac{20}{53} a^{28} - \frac{18}{53} a^{27} + \frac{14}{53} a^{26} - \frac{24}{53} a^{25} - \frac{4}{53} a^{24} - \frac{16}{53} a^{23} + \frac{2}{53} a^{22} - \frac{3}{53} a^{21} + \frac{3}{53} a^{20} + \frac{24}{53} a^{19} + \frac{9}{53} a^{18} + \frac{8}{53} a^{17} + \frac{13}{53} a^{16} + \frac{6}{53} a^{15} - \frac{17}{53} a^{14} - \frac{23}{53} a^{13} - \frac{20}{53} a^{12} - \frac{13}{53} a^{11} + \frac{13}{53} a^{10} + \frac{16}{53} a^{9} + \frac{26}{53} a^{8} + \frac{1}{53} a^{7} - \frac{22}{53} a^{6} - \frac{11}{53} a^{5} + \frac{13}{53} a^{4} - \frac{12}{53} a^{3} - \frac{15}{53} a^{2} + \frac{14}{53} a$, $\frac{1}{53} a^{32} - \frac{6}{53} a^{28} + \frac{20}{53} a^{27} + \frac{5}{53} a^{26} - \frac{12}{53} a^{25} + \frac{2}{53} a^{24} + \frac{23}{53} a^{23} - \frac{10}{53} a^{22} - \frac{14}{53} a^{21} + \frac{15}{53} a^{20} + \frac{22}{53} a^{19} - \frac{18}{53} a^{18} + \frac{24}{53} a^{17} - \frac{3}{53} a^{16} - \frac{21}{53} a^{15} + \frac{3}{53} a^{14} - \frac{5}{53} a^{13} + \frac{21}{53} a^{11} + \frac{26}{53} a^{10} - \frac{8}{53} a^{9} - \frac{25}{53} a^{8} - \frac{6}{53} a^{7} + \frac{5}{53} a^{6} - \frac{5}{53} a^{5} + \frac{9}{53} a^{4} + \frac{14}{53} a^{3} + \frac{5}{53} a^{2} - \frac{15}{53} a$, $\frac{1}{53} a^{33} + \frac{13}{53} a^{28} - \frac{25}{53} a^{27} + \frac{2}{53} a^{26} - \frac{11}{53} a^{25} - \frac{14}{53} a^{24} - \frac{9}{53} a^{23} + \frac{21}{53} a^{22} - \frac{6}{53} a^{21} + \frac{14}{53} a^{20} + \frac{23}{53} a^{19} - \frac{5}{53} a^{18} - \frac{5}{53} a^{17} + \frac{24}{53} a^{16} + \frac{23}{53} a^{15} + \frac{24}{53} a^{14} - \frac{22}{53} a^{13} + \frac{9}{53} a^{12} - \frac{14}{53} a^{11} - \frac{5}{53} a^{10} - \frac{14}{53} a^{9} + \frac{18}{53} a^{8} - \frac{22}{53} a^{7} + \frac{21}{53} a^{6} - \frac{7}{53} a^{5} + \frac{15}{53} a^{4} + \frac{19}{53} a^{3} - \frac{23}{53} a^{2} + \frac{22}{53} a$, $\frac{1}{53} a^{34} - \frac{1}{53} a^{28} + \frac{14}{53} a^{27} - \frac{6}{53} a^{26} + \frac{23}{53} a^{25} - \frac{26}{53} a^{24} + \frac{10}{53} a^{23} - \frac{20}{53} a^{22} - \frac{20}{53} a^{21} + \frac{5}{53} a^{20} + \frac{21}{53} a^{19} - \frac{4}{53} a^{18} - \frac{7}{53} a^{17} + \frac{5}{53} a^{16} + \frac{16}{53} a^{15} - \frac{23}{53} a^{14} - \frac{14}{53} a^{13} + \frac{12}{53} a^{12} + \frac{11}{53} a^{11} + \frac{6}{53} a^{10} + \frac{3}{53} a^{9} - \frac{21}{53} a^{8} + \frac{25}{53} a^{6} - \frac{21}{53} a^{5} + \frac{8}{53} a^{4} - \frac{18}{53} a^{3} + \frac{4}{53} a^{2} + \frac{23}{53} a$, $\frac{1}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{35} + \frac{31159463257131232841872974992888373260142679354560330837602659449300925272042707139143147184381495032304557132425596779175}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{34} + \frac{49924855913287656503510597120324359027152387579654709732190730379736413045969394778824590176731124752517812055850246425017}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{33} - \frac{72476050507457988758586252110913070794045539977020980189985934487710285139602420436084484741792398889258019064381914681216}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{32} + \frac{30976033733098853174205322169340404346679655779511800531689643115293183306542209058462572247546296696249385195975534809635}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{31} + \frac{49563400276673919428590911467836257508443335837113675570676059061768624185288319828966832964263766164165087780385342873032}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{30} - \frac{28792307102208632507471935501215108078594986016356200870444201387527361768546107332304570402302560050351594536180865077066}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{29} + \frac{2245077005342651320181318361182394016405128491530339070891947757066297550653413243591348596568782472225078772255422257832421}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{28} + \frac{954667925198164843662640355089768951798740515778698785161092836876757033559783127859581433064736857977052890543190452208895}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{27} - \frac{790533262100073195345693730264684800294766818278026892039121242361506624966720913702887969851107989606384531027478288382628}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{26} + \frac{2917553660378671569905865591198837743507774013055322088964753977921392862846565409061554219235488848575186032941174183499600}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{25} - \frac{639795270832124815021715283304901367938091983505597587144899017672008570282193629808013656710617664989068806755342334276973}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{24} - \frac{2491065441580978298113945362175532741011086048742982027453132176294924146960582594051833167924029270696614519070169205818790}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{23} - \frac{3220627355412331382689142156732103191957819722894775442812132622909380612583288357031945002820011018556700482426996603375533}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{22} + \frac{2206747738441574587432806457807658314332410444226817881866119890948360857503618592005911548535833543059601716213659620759589}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{21} - \frac{3903869065059565764303041544013009005312017882875005074056773250578212097010186019158336747782166131166584075366134102759729}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{20} + \frac{3740120732679010011536002342907317940277464312181693477399428930924306469373950529140736094756416556371701961660629785440595}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{19} + \frac{1918720834993183215379078791787799211388514273330856793830130559674327743783672409728939763263484247254440289555078804571846}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{18} - \frac{2988735698494336058251062785388150900886045504165806844492897126327176311970961236875665631582228179884103418423921993717158}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{17} + \frac{472373360688849086242825009756794923330866916867185395430961761963059266751881303927598176955270590042257605991566679022052}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{16} + \frac{2787072400237889846017935200424779395451214742050522050494137587049511782803621237021254462525856949276958791409984285323226}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{15} - \frac{3699424290914956677069742485828559970342277626956363470575261670489774394112454970419262711720087366930323756263411206801030}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{14} - \frac{1253237013221812603170154017966861313650819733308156852113295993335386116042693786030499789794830314613539400915691344897186}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{13} - \frac{3347365150070629635258403526284245179827469966411045237847380565421699735523330728700636856596461149757601247775359113186231}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{12} + \frac{1230977052951797116756625348692273768102923148963646786689683584026181546946052374836515283181418455485659730461794923684673}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{11} + \frac{1574411611599810169414293417316499683287625319203409138347193515357580746420855230438524717248056071000197845834621446112701}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{10} + \frac{1376257282119136622845990251453454649092738972284742976562421439132803685809969773324271553801705285651439655558053111011125}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{9} + \frac{2979808559537708529509552858053204889979413753090993616322527675585831071809943760257288974224372215372510166061806024368914}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{8} + \frac{1956072092069136204589952332782909811777105662656825974461096810889271171233149502623215229916789930495623395265270241581323}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{7} - \frac{3619518191161537691006016410613117064778722117250953526551232855303138290245963362493981654586605624855713850414612890539156}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{6} + \frac{2239055446413061855132292557173629447843997981719937592080082983521912348481596358240551679547706265301718582161643424088516}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{5} - \frac{2162995409287684249978048693501698522899641611373604704499756383656017595982568121359177733193316178923991223333406617968586}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{4} + \frac{3853205747875579113468344153298783718173586444205040865428536479944853690430215809383258345796531847912195695030816668555398}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{3} - \frac{1491679259933302137958050398216004737839276842519266929835221099589637912482880600942884817322663000233096646255742341849662}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a^{2} + \frac{1535477242155318050898309099840111275112616725680880727537201081253104574733162116069869681285684096857286713417775897810834}{7808454213148172772319199986796320445289662130161619098460932337559255426828546226318529948803444075267306726756183245189421} a + \frac{34347337874443997739278585230725802599453824383339541716289603679849125057380096740676594270853258398858339159301636621509}{147329324776380618345645282769741895194144568493615454687942119576589725034500872194689244317046114627685032580305344248857}$

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:  $35$
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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{9})^+\), 3.3.13689.1, 3.3.169.1, 3.3.13689.2, 4.4.107653.1, 6.6.14414517.1, 6.6.2436053373.1, \(\Q(\zeta_{13})^+\), 6.6.2436053373.2, 9.9.2565164201769.1, 12.12.53705465637821473811517.1, 12.12.9076223692791829074146373.1, 12.12.210845878198059013.1, 12.12.9076223692791829074146373.2, 18.18.14456408038335708501176406117.1

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 ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{36}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{3}$

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
3Data not computed
$7$7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
7.12.6.2$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
13Data not computed