↑

Properties

Label 40.0.10038339744...5625.1
Degree $40$
Signature $[0, 20]$
Discriminant $3^{20}\cdot 5^{30}\cdot 11^{36}$
Root discriminant $50.12$
Ramified primes $3, 5, 11$
Class number $8420$ (GRH)
Class group $[2, 4210]$ (GRH)
Galois group $C_2\times C_{20}$ (as 40T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, 132, -1397, 14707, -26432, 86679, -154704, 355025, -264572, 869342, -558855, 2360500, -1836384, 3317622, -2519131, 3371093, -1938861, 2342802, -1163338, 1330759, -595562, 659494, -263693, 285912, -95699, 102597, -30778, 32603, -8833, 9197, -2167, 2233, -429, 439, -76, 77, -11, 11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - x^39 + 11*x^38 - 11*x^37 + 77*x^36 - 76*x^35 + 439*x^34 - 429*x^33 + 2233*x^32 - 2167*x^31 + 9197*x^30 - 8833*x^29 + 32603*x^28 - 30778*x^27 + 102597*x^26 - 95699*x^25 + 285912*x^24 - 263693*x^23 + 659494*x^22 - 595562*x^21 + 1330759*x^20 - 1163338*x^19 + 2342802*x^18 - 1938861*x^17 + 3371093*x^16 - 2519131*x^15 + 3317622*x^14 - 1836384*x^13 + 2360500*x^12 - 558855*x^11 + 869342*x^10 - 264572*x^9 + 355025*x^8 - 154704*x^7 + 86679*x^6 - 26432*x^5 + 14707*x^4 - 1397*x^3 + 132*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^40 - x^39 + 11*x^38 - 11*x^37 + 77*x^36 - 76*x^35 + 439*x^34 - 429*x^33 + 2233*x^32 - 2167*x^31 + 9197*x^30 - 8833*x^29 + 32603*x^28 - 30778*x^27 + 102597*x^26 - 95699*x^25 + 285912*x^24 - 263693*x^23 + 659494*x^22 - 595562*x^21 + 1330759*x^20 - 1163338*x^19 + 2342802*x^18 - 1938861*x^17 + 3371093*x^16 - 2519131*x^15 + 3317622*x^14 - 1836384*x^13 + 2360500*x^12 - 558855*x^11 + 869342*x^10 - 264572*x^9 + 355025*x^8 - 154704*x^7 + 86679*x^6 - 26432*x^5 + 14707*x^4 - 1397*x^3 + 132*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{40} - x^{39} + 11 x^{38} - 11 x^{37} + 77 x^{36} - 76 x^{35} + 439 x^{34} - 429 x^{33} + 2233 x^{32} - 2167 x^{31} + 9197 x^{30} - 8833 x^{29} + 32603 x^{28} - 30778 x^{27} + 102597 x^{26} - 95699 x^{25} + 285912 x^{24} - 263693 x^{23} + 659494 x^{22} - 595562 x^{21} + 1330759 x^{20} - 1163338 x^{19} + 2342802 x^{18} - 1938861 x^{17} + 3371093 x^{16} - 2519131 x^{15} + 3317622 x^{14} - 1836384 x^{13} + 2360500 x^{12} - 558855 x^{11} + 869342 x^{10} - 264572 x^{9} + 355025 x^{8} - 154704 x^{7} + 86679 x^{6} - 26432 x^{5} + 14707 x^{4} - 1397 x^{3} + 132 x^{2} - 12 x + 1 \)

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

Invariants

Degree:  $40$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 20]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(100383397447978918530459891214693626269465146363712847232818603515625=3^{20}\cdot 5^{30}\cdot 11^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.12$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(165=3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(131,·)$, $\chi_{165}(4,·)$, $\chi_{165}(133,·)$, $\chi_{165}(134,·)$, $\chi_{165}(8,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(148,·)$, $\chi_{165}(149,·)$, $\chi_{165}(157,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(161,·)$, $\chi_{165}(34,·)$, $\chi_{165}(163,·)$, $\chi_{165}(164,·)$, $\chi_{165}(37,·)$, $\chi_{165}(41,·)$, $\chi_{165}(29,·)$, $\chi_{165}(49,·)$, $\chi_{165}(58,·)$, $\chi_{165}(62,·)$, $\chi_{165}(64,·)$, $\chi_{165}(67,·)$, $\chi_{165}(68,·)$, $\chi_{165}(74,·)$, $\chi_{165}(82,·)$, $\chi_{165}(83,·)$, $\chi_{165}(91,·)$, $\chi_{165}(97,·)$, $\chi_{165}(98,·)$, $\chi_{165}(101,·)$, $\chi_{165}(103,·)$, $\chi_{165}(107,·)$, $\chi_{165}(116,·)$, $\chi_{165}(136,·)$, $\chi_{165}(124,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{629634570480290540834950639222275992422799} a^{37} + \frac{118316768697740837733017598178270793886931}{629634570480290540834950639222275992422799} a^{36} - \frac{51734807124076211868941391443608790673261}{629634570480290540834950639222275992422799} a^{35} - \frac{24366646859096492470783905218235255303017}{629634570480290540834950639222275992422799} a^{34} + \frac{60551692115452210276595333342579295016884}{629634570480290540834950639222275992422799} a^{33} - \frac{267231078312385830737259099983125482708164}{629634570480290540834950639222275992422799} a^{32} + \frac{44457271784143232557312651007509480934698}{629634570480290540834950639222275992422799} a^{31} + \frac{141063812212376680021540514672394965430263}{629634570480290540834950639222275992422799} a^{30} - \frac{106434498063027614228366031843510202825804}{629634570480290540834950639222275992422799} a^{29} - \frac{110861270287380289569587654733350402603797}{629634570480290540834950639222275992422799} a^{28} - \frac{228159209413066943623294312716016371502919}{629634570480290540834950639222275992422799} a^{27} - \frac{50878606598666712680371611523882276124006}{629634570480290540834950639222275992422799} a^{26} - \frac{118534765306822127621542522107489586842277}{629634570480290540834950639222275992422799} a^{25} + \frac{86908930649037492565928511127820578323985}{629634570480290540834950639222275992422799} a^{24} - \frac{4512395809643068406732201046283203403604}{629634570480290540834950639222275992422799} a^{23} + \frac{131858632837866122299444861484614152223893}{629634570480290540834950639222275992422799} a^{22} - \frac{303528682791494631814708954822379165846192}{629634570480290540834950639222275992422799} a^{21} + \frac{22597105270417701179907890224178596116710}{629634570480290540834950639222275992422799} a^{20} - \frac{95259681050252448165173800619866186002839}{629634570480290540834950639222275992422799} a^{19} + \frac{92652582865668493966714086999182976040926}{629634570480290540834950639222275992422799} a^{18} - \frac{305358896986321524753296288024516441817451}{629634570480290540834950639222275992422799} a^{17} + \frac{238389818648032178936260160909448900077047}{629634570480290540834950639222275992422799} a^{16} + \frac{56914081073721928226804519905290746021029}{629634570480290540834950639222275992422799} a^{15} + \frac{148509986471899338979402960049194338469171}{629634570480290540834950639222275992422799} a^{14} + \frac{161721940844068269929196907932573066761898}{629634570480290540834950639222275992422799} a^{13} + \frac{303754880130310890480902168534048353854956}{629634570480290540834950639222275992422799} a^{12} + \frac{287005078842555883942006360389103829034262}{629634570480290540834950639222275992422799} a^{11} - \frac{129990930162086768186009765749014732017164}{629634570480290540834950639222275992422799} a^{10} + \frac{131720431516432040944480164779425469403822}{629634570480290540834950639222275992422799} a^{9} + \frac{193496967657210135056363371959031297099036}{629634570480290540834950639222275992422799} a^{8} + \frac{162137018870024643552584617462055399778811}{629634570480290540834950639222275992422799} a^{7} - \frac{281889675017694803017757408102540501730087}{629634570480290540834950639222275992422799} a^{6} - \frac{265103584644846694491415890329928584489577}{629634570480290540834950639222275992422799} a^{5} - \frac{268650295475026976387569706528580839068977}{629634570480290540834950639222275992422799} a^{4} + \frac{86437515373337786764807914852968433708480}{629634570480290540834950639222275992422799} a^{3} + \frac{19232349253188655237368660160532857877940}{629634570480290540834950639222275992422799} a^{2} - \frac{239450875944267331278974485564290907301405}{629634570480290540834950639222275992422799} a - \frac{875500534847538660340917048907539294928}{629634570480290540834950639222275992422799}$, $\frac{1}{629634570480290540834950639222275992422799} a^{38} - \frac{279189301631482544795622366621145165161574}{629634570480290540834950639222275992422799} a^{36} - \frac{166629701107924840513468995597584317308363}{629634570480290540834950639222275992422799} a^{35} - \frac{106725033285738444102952113724763364616215}{629634570480290540834950639222275992422799} a^{34} + \frac{225640295249646496871226029576569804143999}{629634570480290540834950639222275992422799} a^{33} - \frac{223068363003120649154200625642977607279414}{629634570480290540834950639222275992422799} a^{32} - \frac{224306017561364156770625819702246288480342}{629634570480290540834950639222275992422799} a^{31} - \frac{95218457042463387228076931567289918613812}{629634570480290540834950639222275992422799} a^{30} + \frac{112059727963007762305398889528407937467449}{629634570480290540834950639222275992422799} a^{29} - \frac{37387754176811462600341332478013422251869}{629634570480290540834950639222275992422799} a^{28} - \frac{137016153365677679504915466362223169043064}{629634570480290540834950639222275992422799} a^{27} - \frac{67977261313066489492414257512470753635184}{629634570480290540834950639222275992422799} a^{26} + \frac{72619145862255354264251268331664309443416}{629634570480290540834950639222275992422799} a^{25} - \frac{65688419089101670441507947414413176101872}{629634570480290540834950639222275992422799} a^{24} - \frac{2394421915976823765408558946447040534359}{4806370767025118632327867474979206049029} a^{23} - \frac{120065026190031967037194865001207612427500}{629634570480290540834950639222275992422799} a^{22} - \frac{62103541162156007386801632912769946044027}{629634570480290540834950639222275992422799} a^{21} + \frac{15384533899247132272318088349040094842402}{629634570480290540834950639222275992422799} a^{20} + \frac{19153473444984740241419430188309064887906}{629634570480290540834950639222275992422799} a^{19} + \frac{262208480273953315895189626755171203189777}{629634570480290540834950639222275992422799} a^{18} - \frac{303256350879632560621552885946434149730974}{629634570480290540834950639222275992422799} a^{17} - \frac{81927198568840849895644450883286097416940}{629634570480290540834950639222275992422799} a^{16} - \frac{73925577948816387465317672142618819839231}{629634570480290540834950639222275992422799} a^{15} - \frac{8480542315192368804844829673228243942002}{629634570480290540834950639222275992422799} a^{14} - \frac{14347968307261139702084170908134403668932}{629634570480290540834950639222275992422799} a^{13} - \frac{186545567215499551558817357145174756814638}{629634570480290540834950639222275992422799} a^{12} + \frac{75627534788440856813464565869261639670978}{629634570480290540834950639222275992422799} a^{11} + \frac{277733941026401971896172294283865459131091}{629634570480290540834950639222275992422799} a^{10} + \frac{86986954713531309023911234161304156445959}{629634570480290540834950639222275992422799} a^{9} + \frac{8456159783606142785797965785656993301290}{629634570480290540834950639222275992422799} a^{8} + \frac{35285101202397565400483923678019109872717}{629634570480290540834950639222275992422799} a^{7} + \frac{245558572729382384909466560984958243118178}{629634570480290540834950639222275992422799} a^{6} + \frac{62924705508682707858664656205386068331463}{629634570480290540834950639222275992422799} a^{5} + \frac{114059166565161129617108286123659992989257}{629634570480290540834950639222275992422799} a^{4} - \frac{98120943069512314421082390229828389433656}{629634570480290540834950639222275992422799} a^{3} + \frac{204133931503958538207481391735620410798928}{629634570480290540834950639222275992422799} a^{2} - \frac{184114565099973991298158555815001103932491}{629634570480290540834950639222275992422799} a - \frac{281498466726801882500316454170213302881524}{629634570480290540834950639222275992422799}$, $\frac{1}{629634570480290540834950639222275992422799} a^{39} - \frac{71331589466340306037303009033273866830495}{629634570480290540834950639222275992422799} a^{36} + \frac{136105443283116194699102112599397051816469}{629634570480290540834950639222275992422799} a^{35} - \frac{50123471470280594222648500226690410431059}{629634570480290540834950639222275992422799} a^{34} + \frac{237890735153697060020221960148815585135638}{629634570480290540834950639222275992422799} a^{33} + \frac{91300923910167068197384557433317143433787}{629634570480290540834950639222275992422799} a^{32} + \frac{168597586843640615895593112171836122802003}{629634570480290540834950639222275992422799} a^{31} - \frac{145576703706156821326685860865751062169644}{629634570480290540834950639222275992422799} a^{30} + \frac{203307237520579106431753611024732488152133}{629634570480290540834950639222275992422799} a^{29} + \frac{111856858944649909143821735786169009338162}{629634570480290540834950639222275992422799} a^{28} + \frac{313095271369808635609474297145504855134318}{629634570480290540834950639222275992422799} a^{27} - \frac{214484537245834532764050131184880590357161}{629634570480290540834950639222275992422799} a^{26} - \frac{74751856259404568218563235665008403447210}{629634570480290540834950639222275992422799} a^{25} - \frac{231786609097759983817965816281027788065100}{629634570480290540834950639222275992422799} a^{24} + \frac{16713403102329037280691027146873727640924}{629634570480290540834950639222275992422799} a^{23} - \frac{100082207642136203243156439541143531876497}{629634570480290540834950639222275992422799} a^{22} - \frac{301234304139599288430786035472822864239238}{629634570480290540834950639222275992422799} a^{21} + \frac{283019068865362578408537755507851609112550}{629634570480290540834950639222275992422799} a^{20} - \frac{35608440284084274915931768074061300671102}{629634570480290540834950639222275992422799} a^{19} - \frac{248532624723362743928772479512812479807007}{629634570480290540834950639222275992422799} a^{18} + \frac{27851052494289495148548510221383754315826}{629634570480290540834950639222275992422799} a^{17} - \frac{13491619523071794316028783945484823535262}{629634570480290540834950639222275992422799} a^{16} + \frac{144684494430232327638743443442171281175732}{629634570480290540834950639222275992422799} a^{15} - \frac{32437253544660222573084505317292430410503}{629634570480290540834950639222275992422799} a^{14} + \frac{292220229248757040557250656870047667203430}{629634570480290540834950639222275992422799} a^{13} - \frac{184108425852520360512845533838859072307417}{629634570480290540834950639222275992422799} a^{12} + \frac{272917465703807796040227319636590893568052}{629634570480290540834950639222275992422799} a^{11} - \frac{264030277265544516281043277502067814867102}{629634570480290540834950639222275992422799} a^{10} + \frac{263285334864427018591220414876254218100008}{629634570480290540834950639222275992422799} a^{9} + \frac{41706355698652733760423962019926799847927}{629634570480290540834950639222275992422799} a^{8} + \frac{238637091713360452922589817011267802143254}{629634570480290540834950639222275992422799} a^{7} + \frac{297430846993401042930949079608502052269176}{629634570480290540834950639222275992422799} a^{6} - \frac{70139084003670243965373005144371039701393}{629634570480290540834950639222275992422799} a^{5} + \frac{124039396683073216431453503662191071653688}{629634570480290540834950639222275992422799} a^{4} - \frac{308528238463951576005110362482453205042320}{629634570480290540834950639222275992422799} a^{3} - \frac{50518964670359601515682401828824613683309}{629634570480290540834950639222275992422799} a^{2} - \frac{196825201312434192902621239169311019870644}{629634570480290540834950639222275992422799} a + \frac{122203187733984361454051578366005851753337}{629634570480290540834950639222275992422799}$

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

Class group and class number

$C_{2}\times C_{4210}$, which has order $8420$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $19$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{59880346098408791098299272789362685357516}{629634570480290540834950639222275992422799} a^{39} + \frac{59923417166230540017835647848862686469485}{629634570480290540834950639222275992422799} a^{38} - \frac{658684870851470608554420434033240777735632}{629634570480290540834950639222275992422799} a^{37} + \frac{659102621621900436516916348162576903091230}{629634570480290540834950639222275992422799} a^{36} - \frac{4610794095960294259880943038232685444149424}{629634570480290540834950639222275992422799} a^{35} + \frac{4553629146411418780622486716850854805405508}{629634570480290540834950639222275992422799} a^{34} - \frac{26287470417488836793001176699332133647536284}{629634570480290540834950639222275992422799} a^{33} + \frac{25703507135233631324083436411982875192969514}{629634570480290540834950639222275992422799} a^{32} - \frac{133712600038636776607848997038983559225363077}{629634570480290540834950639222275992422799} a^{31} + \frac{129834014949123602677921755321547272621534533}{629634570480290540834950639222275992422799} a^{30} - \frac{550717774079340817421172323929002886189816475}{629634570480290540834950639222275992422799} a^{29} + \frac{529203987224656135713637988018602321394953229}{629634570480290540834950639222275992422799} a^{28} - \frac{1952266796876913291595389085116591016476079057}{629634570480290540834950639222275992422799} a^{27} + \frac{1843935227846960699017541795713366892238970523}{629634570480290540834950639222275992422799} a^{26} - \frac{6143477561909504249484659739055067873592457650}{629634570480290540834950639222275992422799} a^{25} + \frac{5733290415949261471238810642039932716657538655}{629634570480290540834950639222275992422799} a^{24} - \frac{17120253785234335381878357574589049629028461427}{629634570480290540834950639222275992422799} a^{23} + \frac{15797333989026247915531403973640215763445528824}{629634570480290540834950639222275992422799} a^{22} - \frac{39489889866686315408936674624883437413642872819}{629634570480290540834950639222275992422799} a^{21} + \frac{35677141842047483236327771644652129762158219934}{629634570480290540834950639222275992422799} a^{20} - \frac{79683818001439454646881866492165583462781850252}{629634570480290540834950639222275992422799} a^{19} + \frac{69687756671595218894154698639152661628431092973}{629634570480290540834950639222275992422799} a^{18} - \frac{140281329543899637550028426487498901407507488892}{629634570480290540834950639222275992422799} a^{17} + \frac{116141065724587239923898970013023413143451445866}{629634570480290540834950639222275992422799} a^{16} - \frac{201846595116391300426723804454894605012413800413}{629634570480290540834950639222275992422799} a^{15} + \frac{150892754544876901405905018985861207897128250023}{629634570480290540834950639222275992422799} a^{14} - \frac{198627743652997438843782079881948397446261793932}{629634570480290540834950639222275992422799} a^{13} + \frac{109978283778736209799638562652239529010803785905}{629634570480290540834950639222275992422799} a^{12} - \frac{141292409726405291153449228629236179962201937439}{629634570480290540834950639222275992422799} a^{11} + \frac{33474063152214351307038059166616931237778466293}{629634570480290540834950639222275992422799} a^{10} - \frac{51996724578570492517080933859754757404097951946}{629634570480290540834950639222275992422799} a^{9} + \frac{15846430208429339102104373485823256018140980561}{629634570480290540834950639222275992422799} a^{8} - \frac{21252836162408427135483941509097837595973280506}{629634570480290540834950639222275992422799} a^{7} + \frac{9261548468520440715248056576602962396089895433}{629634570480290540834950639222275992422799} a^{6} - \frac{39606584930697257627771268775517217532247550}{4806370767025118632327867474979206049029} a^{5} + \frac{1573152445923325012601706492198424947717742599}{629634570480290540834950639222275992422799} a^{4} - \frac{6719164888834533842323566636973089673397143}{4806370767025118632327867474979206049029} a^{3} + \frac{83610104176961074416576288145789713275403364}{629634570480290540834950639222275992422799} a^{2} - \frac{7900150876926620030386637623455353375417814}{629634570480290540834950639222275992422799} a + \frac{718191536651028329767565181901402114269030}{629634570480290540834950639222275992422799} \) (order $10$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 276594299286030.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{20}$ (as 40T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2\times C_{20}$
Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{33}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{5}, \sqrt{33})\), 4.0.136125.2, \(\Q(\zeta_{5})\), \(\Q(\zeta_{11})^+\), 8.0.18530015625.3, \(\Q(\zeta_{33})^+\), 10.10.669871503125.1, 10.10.1790566527853125.1, 20.20.3206128490667995866421572265625.1, 20.0.10019151533337487082567413330078125.1, 20.0.1402274470934209014892578125.1

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 $20^{2}$ R R $20^{2}$ R $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$ $20^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$ $20^{2}$ $20^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

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
5Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$