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Properties

Label 46.46.4452622216...8125.1
Degree $46$
Signature $[46, 0]$
Discriminant $5^{23}\cdot 47^{44}$
Root discriminant $88.90$
Ramified primes $5, 47$
Class number Not computed
Class group Not computed
Galois group $C_{46}$ (as 46T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -12, 342, 352, -19657, -6578, 445588, 114257, -5308732, -1297868, 38414233, 9203272, -183867242, -43115073, 615043968, 140457727, -1492308292, -329962748, 2694891598, 573064052, -3686881837, -748176098, 3866596078, 742002577, -3130570732, -562037616, 1962978291, 325518500, -952979825, -143761015, 356957890, 48099584, -102471229, -12065268, 22311838, 2234528, -3629323, -298794, 431244, 27871, -36151, -1712, 2017, 62, -67, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^46 - x^45 - 67*x^44 + 62*x^43 + 2017*x^42 - 1712*x^41 - 36151*x^40 + 27871*x^39 + 431244*x^38 - 298794*x^37 - 3629323*x^36 + 2234528*x^35 + 22311838*x^34 - 12065268*x^33 - 102471229*x^32 + 48099584*x^31 + 356957890*x^30 - 143761015*x^29 - 952979825*x^28 + 325518500*x^27 + 1962978291*x^26 - 562037616*x^25 - 3130570732*x^24 + 742002577*x^23 + 3866596078*x^22 - 748176098*x^21 - 3686881837*x^20 + 573064052*x^19 + 2694891598*x^18 - 329962748*x^17 - 1492308292*x^16 + 140457727*x^15 + 615043968*x^14 - 43115073*x^13 - 183867242*x^12 + 9203272*x^11 + 38414233*x^10 - 1297868*x^9 - 5308732*x^8 + 114257*x^7 + 445588*x^6 - 6578*x^5 - 19657*x^4 + 352*x^3 + 342*x^2 - 12*x - 1)
 
gp: K = bnfinit(x^46 - x^45 - 67*x^44 + 62*x^43 + 2017*x^42 - 1712*x^41 - 36151*x^40 + 27871*x^39 + 431244*x^38 - 298794*x^37 - 3629323*x^36 + 2234528*x^35 + 22311838*x^34 - 12065268*x^33 - 102471229*x^32 + 48099584*x^31 + 356957890*x^30 - 143761015*x^29 - 952979825*x^28 + 325518500*x^27 + 1962978291*x^26 - 562037616*x^25 - 3130570732*x^24 + 742002577*x^23 + 3866596078*x^22 - 748176098*x^21 - 3686881837*x^20 + 573064052*x^19 + 2694891598*x^18 - 329962748*x^17 - 1492308292*x^16 + 140457727*x^15 + 615043968*x^14 - 43115073*x^13 - 183867242*x^12 + 9203272*x^11 + 38414233*x^10 - 1297868*x^9 - 5308732*x^8 + 114257*x^7 + 445588*x^6 - 6578*x^5 - 19657*x^4 + 352*x^3 + 342*x^2 - 12*x - 1, 1)
 

Normalized defining polynomial

\( x^{46} - x^{45} - 67 x^{44} + 62 x^{43} + 2017 x^{42} - 1712 x^{41} - 36151 x^{40} + 27871 x^{39} + 431244 x^{38} - 298794 x^{37} - 3629323 x^{36} + 2234528 x^{35} + 22311838 x^{34} - 12065268 x^{33} - 102471229 x^{32} + 48099584 x^{31} + 356957890 x^{30} - 143761015 x^{29} - 952979825 x^{28} + 325518500 x^{27} + 1962978291 x^{26} - 562037616 x^{25} - 3130570732 x^{24} + 742002577 x^{23} + 3866596078 x^{22} - 748176098 x^{21} - 3686881837 x^{20} + 573064052 x^{19} + 2694891598 x^{18} - 329962748 x^{17} - 1492308292 x^{16} + 140457727 x^{15} + 615043968 x^{14} - 43115073 x^{13} - 183867242 x^{12} + 9203272 x^{11} + 38414233 x^{10} - 1297868 x^{9} - 5308732 x^{8} + 114257 x^{7} + 445588 x^{6} - 6578 x^{5} - 19657 x^{4} + 352 x^{3} + 342 x^{2} - 12 x - 1 \)

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

Invariants

Degree:  $46$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[46, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(445262221645814097378614331350194306504897709623466822770698795048558574688434600830078125=5^{23}\cdot 47^{44}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.90$
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:  $5, 47$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(235=5\cdot 47\)
Dirichlet character group:    $\lbrace$$\chi_{235}(1,·)$, $\chi_{235}(131,·)$, $\chi_{235}(4,·)$, $\chi_{235}(6,·)$, $\chi_{235}(136,·)$, $\chi_{235}(9,·)$, $\chi_{235}(14,·)$, $\chi_{235}(16,·)$, $\chi_{235}(21,·)$, $\chi_{235}(24,·)$, $\chi_{235}(159,·)$, $\chi_{235}(34,·)$, $\chi_{235}(36,·)$, $\chi_{235}(166,·)$, $\chi_{235}(169,·)$, $\chi_{235}(49,·)$, $\chi_{235}(51,·)$, $\chi_{235}(54,·)$, $\chi_{235}(56,·)$, $\chi_{235}(59,·)$, $\chi_{235}(61,·)$, $\chi_{235}(191,·)$, $\chi_{235}(64,·)$, $\chi_{235}(96,·)$, $\chi_{235}(194,·)$, $\chi_{235}(196,·)$, $\chi_{235}(71,·)$, $\chi_{235}(74,·)$, $\chi_{235}(204,·)$, $\chi_{235}(206,·)$, $\chi_{235}(79,·)$, $\chi_{235}(81,·)$, $\chi_{235}(84,·)$, $\chi_{235}(216,·)$, $\chi_{235}(89,·)$, $\chi_{235}(224,·)$, $\chi_{235}(144,·)$, $\chi_{235}(101,·)$, $\chi_{235}(209,·)$, $\chi_{235}(106,·)$, $\chi_{235}(111,·)$, $\chi_{235}(189,·)$, $\chi_{235}(119,·)$, $\chi_{235}(121,·)$, $\chi_{235}(126,·)$, $\chi_{235}(149,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$, $a^{40}$, $a^{41}$, $a^{42}$, $a^{43}$, $\frac{1}{281} a^{44} + \frac{66}{281} a^{43} + \frac{10}{281} a^{42} + \frac{20}{281} a^{41} + \frac{90}{281} a^{40} + \frac{32}{281} a^{39} + \frac{96}{281} a^{38} + \frac{76}{281} a^{37} + \frac{108}{281} a^{36} + \frac{75}{281} a^{35} + \frac{50}{281} a^{34} + \frac{79}{281} a^{33} + \frac{74}{281} a^{32} + \frac{56}{281} a^{31} - \frac{50}{281} a^{30} + \frac{19}{281} a^{29} + \frac{123}{281} a^{28} - \frac{140}{281} a^{27} + \frac{123}{281} a^{26} - \frac{22}{281} a^{25} - \frac{79}{281} a^{24} + \frac{134}{281} a^{23} + \frac{109}{281} a^{22} - \frac{123}{281} a^{21} - \frac{102}{281} a^{20} + \frac{54}{281} a^{19} + \frac{37}{281} a^{18} - \frac{87}{281} a^{17} - \frac{85}{281} a^{16} + \frac{92}{281} a^{15} - \frac{114}{281} a^{14} - \frac{51}{281} a^{13} + \frac{42}{281} a^{12} + \frac{52}{281} a^{11} + \frac{41}{281} a^{10} - \frac{119}{281} a^{9} + \frac{32}{281} a^{8} - \frac{18}{281} a^{7} - \frac{107}{281} a^{6} + \frac{119}{281} a^{5} - \frac{113}{281} a^{4} - \frac{114}{281} a^{3} + \frac{40}{281} a^{2} - \frac{132}{281} a + \frac{67}{281}$, $\frac{1}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{45} + \frac{10253184897780381290018718177793983562191859285100527271253142365252309467200194377287518455322}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{44} - \frac{1964189594456585568816993497554508792291374507406023025746337740138963135047315831708865160380927}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{43} - \frac{1572639905838239488041256920017874563372663523324886131722161785312603755056282399814316893888646}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{42} + \frac{2160885366131095927626111803118461019691638461816719592776409193895991761149292155572479962509312}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{41} + \frac{1556262303886871106343110999011935933890681151058070263120076985325773985968634459984516519246247}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{40} - \frac{1837600400323103959830547871399693312535314351179773931573137854647718734998378551893740032326930}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{39} + \frac{2601913649627420120915473630176397201322165954658500118597369697874335183574639426246318333703969}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{38} - \frac{92300742163680678040785599083916766342178024719155225402248396393575213523786635032107715967763}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{37} - \frac{397017301409333213557274317757563748902452523513078386972453172758568666534489212721860438841196}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{36} + \frac{536155950020905709507302465080965419834591042420565779164869670208283846350111946161161021171765}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{35} - \frac{2691643556790941972832730450508366674301346087325177086184137721303794552976149158515321634520837}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{34} + \frac{2628641672959164888956182979322202152006805315926595108816411757404417768308579259122371385246148}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{33} - \frac{3006174440956338059793759248377735986866214208370031570794034521924140391253254662095556158870918}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{32} - \frac{62052172588154763496378059647660568393034636042043694495099328977765087353272546320383341967142}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{31} + \frac{1260991892457829261408557267658203607476777519663357499535342411438629721115202742361542729116303}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{30} + \frac{196675939885582133766491433855518189554532160906970919055287210574552493971073969644901133701383}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{29} - \frac{2210033569237193052390173835782969181172870420792496146476840243157239203147633971617526054584068}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{28} + \frac{2844050781217126414822257089841162931129537572775864311249766628814206415312051797045977453545671}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{27} + \frac{419956628156432882967895265463298217047163459733105460878780416480596664219651381368139349634845}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{26} + \frac{2442474865477566724033517825698851428435450037053406637987330701503499162281417149836686515602476}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{25} - \frac{1131929226497328460993145919491046827788834329414526261882539141631133227323516466613370052133642}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{24} + \frac{2122010623117531946235543086229881294071644236906029928562119493730275588371828707664861855269513}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{23} + \frac{4238591075604683321656651049817645805884191847728224877695642995409182742687604510512162101982}{22171841413858554043815141269734464405161198755616584356835245083057044257230667771497970648839} a^{22} + \frac{103234837906424429558992165600481755549854159012894764605851712156551771551783737007167919417928}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{21} - \frac{2562638863157511297496715972375009197099701513521851665839808661475958879350907077573121250317418}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{20} + \frac{335492424440993057679877128358359263201817696709915837622774613301614929817247956866013839357186}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{19} + \frac{2594232507660735053797020416157951219609125795794441658677295579829798479156863645829167824428019}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{18} + \frac{1309674455945289622408623108793900740772187392156692937384464488320250385670439241495861293045709}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{17} - \frac{732386368201625318225520747198889359121028308890418022700476906249446034759250188487749336310874}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{16} - \frac{1160595383582106576781518386938822713363668993033616294724490629917661051449359863361987424829448}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{15} + \frac{2400824994050240954211123307125754464507242656158758449234514213464228625107228375312997774809781}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{14} - \frac{2447810862885424698808827423331646334647680586802069807194574686195691622911909045221014975553546}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{13} - \frac{151410784388012486349149014613397188447718555114928895238139720139240867184235117510988962152009}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{12} - \frac{2556604200743431275358673261551445732405021534294719577061909620069433669635321224965166861140828}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{11} - \frac{2535425709997300592041005142486572723143597544048493966104298157802926815221482556253842802341491}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{10} - \frac{1419597266206200504105169641869073360585590553238923598926974787045529052926667153309113230876782}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{9} + \frac{3097809787709023554580917020063478726344083733562325346765886532601768197086978038712562102637554}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{8} - \frac{1666662002537436607475064015060708992014985667767330821599284628617143642187153966216308813397573}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{7} + \frac{788560566033023535435725041913599260288301678088589426431105042781168642205060363842007247786112}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{6} + \frac{787595787153057270372456921884739436807414843268341315291559415180549982063480002472002052624235}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{5} + \frac{141447836320409120333293627771552021152467012511050185838798487573862543899202494587106402660130}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{4} - \frac{1905884211865865996956985770357989648017455403329762510827780573398754571526650200643201215123346}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{3} + \frac{819421249691791800174300911872374939672241493413423913122673273534925313391583641264581804123930}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a^{2} + \frac{2227706020846963473045850912201264861449192031543246507526947612444509393799551128229684677102175}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759} a - \frac{262406488225222425196601517803171396777561948885348772388789875220826074677425328460900720802920}{6230287437294253686312054696795384497850296850328260204270703868339029436281817643790929752323759}$

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:  $45$
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

$C_{46}$ (as 46T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 46
The 46 conjugacy class representatives for $C_{46}$
Character table for $C_{46}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{47})^+\)

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 $46$ $46$ R $46$ $23^{2}$ $46$ $46$ $23^{2}$ $46$ $23^{2}$ $23^{2}$ $46$ $23^{2}$ $46$ R $46$ $23^{2}$

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
5Data not computed
47Data not computed