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Properties

Label 34.34.9531909045...3457.1
Degree $34$
Signature $[34, 0]$
Discriminant $17^{65}$
Root discriminant $225.08$
Ramified prime $17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{34}$ (as 34T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![251, -6664, -100572, 4303737, -25029542, -259612083, 2265179127, 1833018183, -21023817567, -5220519953, 73982920619, 5067071646, -124117063546, 773104733, 110380218272, -3315978418, -58195871911, 2035532841, 19613952028, -643842581, -4421480547, 123620855, 682705397, -15126583, -72718452, 1181483, 5298458, -56712, -257142, 1513, 7871, -17, -136, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^34 - 136*x^32 - 17*x^31 + 7871*x^30 + 1513*x^29 - 257142*x^28 - 56712*x^27 + 5298458*x^26 + 1181483*x^25 - 72718452*x^24 - 15126583*x^23 + 682705397*x^22 + 123620855*x^21 - 4421480547*x^20 - 643842581*x^19 + 19613952028*x^18 + 2035532841*x^17 - 58195871911*x^16 - 3315978418*x^15 + 110380218272*x^14 + 773104733*x^13 - 124117063546*x^12 + 5067071646*x^11 + 73982920619*x^10 - 5220519953*x^9 - 21023817567*x^8 + 1833018183*x^7 + 2265179127*x^6 - 259612083*x^5 - 25029542*x^4 + 4303737*x^3 - 100572*x^2 - 6664*x + 251)
 
gp: K = bnfinit(x^34 - 136*x^32 - 17*x^31 + 7871*x^30 + 1513*x^29 - 257142*x^28 - 56712*x^27 + 5298458*x^26 + 1181483*x^25 - 72718452*x^24 - 15126583*x^23 + 682705397*x^22 + 123620855*x^21 - 4421480547*x^20 - 643842581*x^19 + 19613952028*x^18 + 2035532841*x^17 - 58195871911*x^16 - 3315978418*x^15 + 110380218272*x^14 + 773104733*x^13 - 124117063546*x^12 + 5067071646*x^11 + 73982920619*x^10 - 5220519953*x^9 - 21023817567*x^8 + 1833018183*x^7 + 2265179127*x^6 - 259612083*x^5 - 25029542*x^4 + 4303737*x^3 - 100572*x^2 - 6664*x + 251, 1)
 

Normalized defining polynomial

\( x^{34} - 136 x^{32} - 17 x^{31} + 7871 x^{30} + 1513 x^{29} - 257142 x^{28} - 56712 x^{27} + 5298458 x^{26} + 1181483 x^{25} - 72718452 x^{24} - 15126583 x^{23} + 682705397 x^{22} + 123620855 x^{21} - 4421480547 x^{20} - 643842581 x^{19} + 19613952028 x^{18} + 2035532841 x^{17} - 58195871911 x^{16} - 3315978418 x^{15} + 110380218272 x^{14} + 773104733 x^{13} - 124117063546 x^{12} + 5067071646 x^{11} + 73982920619 x^{10} - 5220519953 x^{9} - 21023817567 x^{8} + 1833018183 x^{7} + 2265179127 x^{6} - 259612083 x^{5} - 25029542 x^{4} + 4303737 x^{3} - 100572 x^{2} - 6664 x + 251 \)

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

Invariants

Degree:  $34$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[34, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(95319090450218007303742536355848761234066170796000792973413605849481890760893457=17^{65}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $225.08$
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:  $17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(289=17^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{289}(256,·)$, $\chi_{289}(1,·)$, $\chi_{289}(135,·)$, $\chi_{289}(137,·)$, $\chi_{289}(271,·)$, $\chi_{289}(16,·)$, $\chi_{289}(273,·)$, $\chi_{289}(18,·)$, $\chi_{289}(152,·)$, $\chi_{289}(154,·)$, $\chi_{289}(288,·)$, $\chi_{289}(33,·)$, $\chi_{289}(35,·)$, $\chi_{289}(169,·)$, $\chi_{289}(171,·)$, $\chi_{289}(50,·)$, $\chi_{289}(52,·)$, $\chi_{289}(186,·)$, $\chi_{289}(188,·)$, $\chi_{289}(67,·)$, $\chi_{289}(69,·)$, $\chi_{289}(203,·)$, $\chi_{289}(205,·)$, $\chi_{289}(84,·)$, $\chi_{289}(86,·)$, $\chi_{289}(220,·)$, $\chi_{289}(222,·)$, $\chi_{289}(101,·)$, $\chi_{289}(103,·)$, $\chi_{289}(237,·)$, $\chi_{289}(239,·)$, $\chi_{289}(118,·)$, $\chi_{289}(120,·)$, $\chi_{289}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $\frac{1}{977} a^{31} - \frac{278}{977} a^{30} + \frac{219}{977} a^{29} - \frac{69}{977} a^{28} - \frac{465}{977} a^{27} + \frac{372}{977} a^{26} + \frac{55}{977} a^{25} - \frac{372}{977} a^{24} - \frac{139}{977} a^{23} - \frac{357}{977} a^{22} - \frac{412}{977} a^{21} + \frac{180}{977} a^{20} + \frac{462}{977} a^{19} + \frac{426}{977} a^{18} - \frac{186}{977} a^{17} - \frac{105}{977} a^{16} - \frac{33}{977} a^{15} - \frac{414}{977} a^{14} - \frac{403}{977} a^{13} - \frac{28}{977} a^{12} + \frac{188}{977} a^{11} - \frac{171}{977} a^{10} - \frac{99}{977} a^{9} + \frac{412}{977} a^{8} + \frac{406}{977} a^{7} - \frac{89}{977} a^{6} + \frac{56}{977} a^{5} - \frac{220}{977} a^{4} + \frac{157}{977} a^{3} + \frac{427}{977} a^{2} - \frac{441}{977} a - \frac{241}{977}$, $\frac{1}{174883} a^{32} - \frac{12}{174883} a^{31} - \frac{58097}{174883} a^{30} + \frac{13243}{174883} a^{29} + \frac{78881}{174883} a^{28} + \frac{2715}{174883} a^{27} + \frac{2284}{174883} a^{26} - \frac{23845}{174883} a^{25} + \frac{25965}{174883} a^{24} + \frac{21289}{174883} a^{23} - \frac{41639}{174883} a^{22} - \frac{26367}{174883} a^{21} - \frac{63036}{174883} a^{20} + \frac{85215}{174883} a^{19} - \frac{30489}{174883} a^{18} + \frac{12947}{174883} a^{17} + \frac{72668}{174883} a^{16} - \frac{48272}{174883} a^{15} + \frac{43839}{174883} a^{14} - \frac{18319}{174883} a^{13} + \frac{74808}{174883} a^{12} - \frac{32231}{174883} a^{11} + \frac{83379}{174883} a^{10} + \frac{62985}{174883} a^{9} - \frac{30690}{174883} a^{8} - \frac{59160}{174883} a^{7} - \frac{82238}{174883} a^{6} - \frac{79116}{174883} a^{5} + \frac{86233}{174883} a^{4} - \frac{57465}{174883} a^{3} + \frac{57452}{174883} a^{2} - \frac{83352}{174883} a + \frac{66812}{174883}$, $\frac{1}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{33} + \frac{876604335804904794316327473143928705555723874476437654017826358214671071342551970725203992591031366932967984611477155745686441999505423}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{32} + \frac{90917977131262572574300098741786868096475870261546682127144241894125294765278301834273287949939680179194862572801797629508464015627639737}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{31} + \frac{17992930310245009690231522709700759135275783494571439301033196741027297937224399074946137899471842380386893570742699246009721245932112854286}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{30} + \frac{143397094351137270780509262602241675972744848365056634910105256972284568700009834720146052288469194846340643116200856711780268274759083916581}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{29} + \frac{4796522101328444899948036295143786942370593683211437732381317981014076077911186492742297019507227999978916671982022563043380003820458584911}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{28} + \frac{108876275889477322428698338513556404000442167490225719558126328090086443568444723798518617415151192478755988111226548718619203461285760421506}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{27} - \frac{164390044250623897247229009849979346251017476243499018949931658836939308541449791969336968914223951527475273838837000072961331782542213459467}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{26} + \frac{36049903026904373702907751123410956517196287986762704622046539282634719431751578137661053692612794142968218319913614836997021022765139133207}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{25} + \frac{54819035566184637639578918879482658659181413780227850153765264172511575257909576008044405215766537910966140493678234307340013999746153385035}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{24} - \frac{165664660842808475174959002727104185394692616270915585729746678582877663515924398885349042509916499954432993906329575796101800796230877986864}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{23} + \frac{134438733685296435330232407839133705366160538810935831861260500558812880788303919762162166145092498839786008856773120901129142128125013720240}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{22} + \frac{12104263527680511558673645789246362879852027692842186188238788673750836624040105908982469059892475930935631355426859890877262042278666486391}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{21} - \frac{59102110448818183061602618892738579212221145271965688661873169741495910436948369227933419071154247490501846785267677271930287364237414362907}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{20} + \frac{155447301112666777575153038723378634338311930596117193814287623282597269192359542157745357003045131686168646797251037967785228846904634123465}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{19} - \frac{48461620349350687044673554220326769141702245885182218952288736123924590051241990334316730684901549368965449516193058629710360752018345328409}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{18} + \frac{189446245206045433436000320798543952800854618110613691986209315725079678397699233373532488313330975600373705131559487692942094216264129151699}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{17} + \frac{91805940555493427956404603536757410232518759047780544196174508581633520640867678777520736239113024556938114302328299756006884279634056104}{433214113776783237230291466678249547649879128602273985808895677987825055116466302608695103746976295737263312576265561057455562995517359307} a^{16} - \frac{210516448154948115486117161886515000949356914016799998436276596215880892305832414054516279900408822148752056531344687916668101008540334963566}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{15} + \frac{35248803728304784827357283540196830892323124082269130275628857235064333383747820792348197753392487279599711308039694648260786555300531608086}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{14} - \frac{128389830312607325119683027947081981127281725310989573675912206709561476014045015900130578924740213390934509944258749307506054737635373139705}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{13} + \frac{102250891481619659973074477893365606818915439544397329809695726545046634557722178505856699356913832505893058080591911504806275345716306911264}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{12} - \frac{86539068343573408666978318452151286885408659499243762600791200546300890934839076959950434002031521472525078342703037299963020064958837278088}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{11} + \frac{161364708850109780466169434553473734389366152720563719770597439532371114738446518271334155932702740721894492517587690691470545602179794447872}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{10} - \frac{36111781913835742249388251662623533752470923082840322725754294928609500839688665629577471746997913812651144068029401591733215051664866608043}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{9} + \frac{18805221811651574891033789961015683076634017061592431560760110265927977217584922624142064366099591683992900717103643460777794954180177044037}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{8} - \frac{25556145273723346161688736133247438445894408404552977127021793695961586082862073892543629356369041450993605598247320713287797460255680169063}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{7} + \frac{171338017747337187654171555682012721269227127512969933098592384088887391745678317743752181770699537456808709374622676598717840940074597371843}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{6} + \frac{67844718692167043945717624106121110014367904256909227597291898086528277909240351590733680952184832716212031804725678615765778204551261996117}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{5} + \frac{121627861575622631627138607770880469778289963727399534409133422639729172453876727716098883966403243357697989849017894869371633914558650389749}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{4} - \frac{65107607582783007957231931720101080063685790840928113826216414747523802018269231094902656597979528100515191931256130484773622870314567389188}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{3} + \frac{25715614441537866391987775714741818690534942198900284289809819293487959077474889370136751452581495012999273010861759786162527136131672315511}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a^{2} + \frac{51384916734214108851262223662893995731238133033389061336126289301516771258793083437696158478204594363565459764572867864097526356069258906930}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939} a + \frac{79068237680411043549222854525205657922205412979844637113447060158105628311833277840906076073491886164542200676658782967384887277164598853067}{423250189159917222773994762944649808053931908644421684135291077394105078848787577648695116360795840935306256387011453153134085046620460042939}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $33$
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:  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:  \( 172609697076036130000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{34}$ (as 34T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 34
The 34 conjugacy class representatives for $C_{34}$
Character table for $C_{34}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 17.17.2367911594760467245844106297320951247361.1

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 $17^{2}$ $34$ $34$ $34$ $34$ $17^{2}$ R $17^{2}$ $34$ $34$ $34$ $34$ $34$ $17^{2}$ $17^{2}$ $17^{2}$ $17^{2}$

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
17Data not computed