Properties

Label 35.35.106...241.1
Degree $35$
Signature $[35, 0]$
Discriminant $1.061\times 10^{79}$
Root discriminant $181.08$
Ramified prime $211$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{35}$ (as 35T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^35 - x^34 - 102*x^33 + 231*x^32 + 4336*x^31 - 15350*x^30 - 93149*x^29 + 492148*x^28 + 885684*x^27 - 8797227*x^26 + 2494886*x^25 + 89830596*x^24 - 154446290*x^23 - 471756827*x^22 + 1664362308*x^21 + 444168124*x^20 - 8565464505*x^19 + 8653560326*x^18 + 19862461891*x^17 - 47903559470*x^16 + 736744022*x^15 + 100720726682*x^14 - 96083394992*x^13 - 62920019542*x^12 + 165504701751*x^11 - 60624002242*x^10 - 87564669300*x^9 + 86642725878*x^8 - 451900172*x^7 - 32473918206*x^6 + 11871839347*x^5 + 3432884529*x^4 - 2696920777*x^3 + 146105313*x^2 + 172726939*x - 27756643)
 
gp: K = bnfinit(x^35 - x^34 - 102*x^33 + 231*x^32 + 4336*x^31 - 15350*x^30 - 93149*x^29 + 492148*x^28 + 885684*x^27 - 8797227*x^26 + 2494886*x^25 + 89830596*x^24 - 154446290*x^23 - 471756827*x^22 + 1664362308*x^21 + 444168124*x^20 - 8565464505*x^19 + 8653560326*x^18 + 19862461891*x^17 - 47903559470*x^16 + 736744022*x^15 + 100720726682*x^14 - 96083394992*x^13 - 62920019542*x^12 + 165504701751*x^11 - 60624002242*x^10 - 87564669300*x^9 + 86642725878*x^8 - 451900172*x^7 - 32473918206*x^6 + 11871839347*x^5 + 3432884529*x^4 - 2696920777*x^3 + 146105313*x^2 + 172726939*x - 27756643, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27756643, 172726939, 146105313, -2696920777, 3432884529, 11871839347, -32473918206, -451900172, 86642725878, -87564669300, -60624002242, 165504701751, -62920019542, -96083394992, 100720726682, 736744022, -47903559470, 19862461891, 8653560326, -8565464505, 444168124, 1664362308, -471756827, -154446290, 89830596, 2494886, -8797227, 885684, 492148, -93149, -15350, 4336, 231, -102, -1, 1]);
 

\( x^{35} - x^{34} - 102 x^{33} + 231 x^{32} + 4336 x^{31} - 15350 x^{30} - 93149 x^{29} + 492148 x^{28} + 885684 x^{27} - 8797227 x^{26} + 2494886 x^{25} + 89830596 x^{24} - 154446290 x^{23} - 471756827 x^{22} + 1664362308 x^{21} + 444168124 x^{20} - 8565464505 x^{19} + 8653560326 x^{18} + 19862461891 x^{17} - 47903559470 x^{16} + 736744022 x^{15} + 100720726682 x^{14} - 96083394992 x^{13} - 62920019542 x^{12} + 165504701751 x^{11} - 60624002242 x^{10} - 87564669300 x^{9} + 86642725878 x^{8} - 451900172 x^{7} - 32473918206 x^{6} + 11871839347 x^{5} + 3432884529 x^{4} - 2696920777 x^{3} + 146105313 x^{2} + 172726939 x - 27756643 \)

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

Invariants

Degree:  $35$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[35, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(106\!\cdots\!241\)\(\medspace = 211^{34}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $181.08$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $211$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $35$
This field is Galois and abelian over $\Q$.
Conductor:  \(211\)
Dirichlet character group:    $\lbrace$$\chi_{211}(1,·)$, $\chi_{211}(5,·)$, $\chi_{211}(65,·)$, $\chi_{211}(11,·)$, $\chi_{211}(13,·)$, $\chi_{211}(143,·)$, $\chi_{211}(144,·)$, $\chi_{211}(148,·)$, $\chi_{211}(151,·)$, $\chi_{211}(25,·)$, $\chi_{211}(169,·)$, $\chi_{211}(171,·)$, $\chi_{211}(55,·)$, $\chi_{211}(184,·)$, $\chi_{211}(58,·)$, $\chi_{211}(188,·)$, $\chi_{211}(64,·)$, $\chi_{211}(193,·)$, $\chi_{211}(203,·)$, $\chi_{211}(71,·)$, $\chi_{211}(183,·)$, $\chi_{211}(76,·)$, $\chi_{211}(79,·)$, $\chi_{211}(82,·)$, $\chi_{211}(87,·)$, $\chi_{211}(199,·)$, $\chi_{211}(96,·)$, $\chi_{211}(107,·)$, $\chi_{211}(109,·)$, $\chi_{211}(113,·)$, $\chi_{211}(114,·)$, $\chi_{211}(121,·)$, $\chi_{211}(122,·)$, $\chi_{211}(123,·)$, $\chi_{211}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $\frac{1}{197} a^{32} + \frac{45}{197} a^{31} - \frac{66}{197} a^{30} + \frac{24}{197} a^{29} + \frac{28}{197} a^{28} + \frac{32}{197} a^{27} + \frac{10}{197} a^{26} - \frac{82}{197} a^{25} - \frac{37}{197} a^{24} - \frac{35}{197} a^{23} - \frac{18}{197} a^{22} - \frac{41}{197} a^{21} - \frac{66}{197} a^{20} - \frac{85}{197} a^{19} + \frac{12}{197} a^{18} + \frac{56}{197} a^{17} - \frac{69}{197} a^{16} - \frac{76}{197} a^{15} + \frac{36}{197} a^{14} + \frac{43}{197} a^{12} - \frac{82}{197} a^{11} + \frac{75}{197} a^{10} + \frac{27}{197} a^{9} + \frac{40}{197} a^{8} + \frac{11}{197} a^{7} - \frac{16}{197} a^{6} + \frac{27}{197} a^{5} - \frac{44}{197} a^{4} - \frac{23}{197} a^{3} + \frac{53}{197} a^{2} - \frac{92}{197} a - \frac{82}{197}$, $\frac{1}{83172241096871} a^{33} - \frac{186918754030}{83172241096871} a^{32} - \frac{29487184447047}{83172241096871} a^{31} + \frac{32429639035710}{83172241096871} a^{30} - \frac{8601971861935}{83172241096871} a^{29} - \frac{65852930420}{197558767451} a^{28} + \frac{26136459732690}{83172241096871} a^{27} - \frac{9684808881901}{83172241096871} a^{26} + \frac{32787029758546}{83172241096871} a^{25} - \frac{7837354950190}{83172241096871} a^{24} + \frac{2577921281611}{83172241096871} a^{23} + \frac{33840948166971}{83172241096871} a^{22} + \frac{5339136920227}{83172241096871} a^{21} - \frac{38623550127827}{83172241096871} a^{20} - \frac{32453801918116}{83172241096871} a^{19} - \frac{18088722387431}{83172241096871} a^{18} - \frac{1076798749129}{83172241096871} a^{17} - \frac{21367964782084}{83172241096871} a^{16} + \frac{4914930418145}{83172241096871} a^{15} - \frac{870518379417}{83172241096871} a^{14} - \frac{8688365443893}{83172241096871} a^{13} - \frac{24990965101303}{83172241096871} a^{12} + \frac{10094400217555}{83172241096871} a^{11} + \frac{9714379157979}{83172241096871} a^{10} - \frac{40592301343324}{83172241096871} a^{9} + \frac{13136019694027}{83172241096871} a^{8} + \frac{5860741616454}{83172241096871} a^{7} + \frac{12350555662586}{83172241096871} a^{6} + \frac{11315216068292}{83172241096871} a^{5} - \frac{9537757002142}{83172241096871} a^{4} - \frac{18817342604566}{83172241096871} a^{3} - \frac{14417941243154}{83172241096871} a^{2} - \frac{21003688906293}{83172241096871} a + \frac{34366965133417}{83172241096871}$, $\frac{1}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{34} - \frac{20963968741638499737557531950053061038166927250128641302807828215785}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{33} - \frac{15367386506170355223823941135229611084390705021885672878080588462948824705724606}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{32} + \frac{1885207604705584758727947376741805391454490888464651812285349842883221189856953804}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{31} - \frac{1039830256667184158322212652927671969860973338497192908371686665937131779119154602}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{30} - \frac{623597928915020564108760580495206376220839443991511709267813360191132259281864782}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{29} + \frac{1512737879870621054570238259727754364716051683557658347471269697848248015753594322}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{28} + \frac{2787969849359295219075712792256017406863778873217689942989615215515202131618700765}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{27} + \frac{1610570400686294150552790049717186181293472548873219611204742269873263876420684557}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{26} - \frac{1960875767836342234936878496080336403964408889697461726684347728684411407075344580}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{25} - \frac{572837678589227019142239213887304074635716776258833779989826252974421861059348656}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{24} - \frac{2160082566117782029728733644285470342886492796750241073946366167249286520401268204}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{23} - \frac{565287994612776057804875973984678668937419135509888352555414520102716417889773213}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{22} + \frac{1399915828467336882860242260340827021884763460776045843313437308814320132678258121}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{21} - \frac{993179540128082647429895721695292616600238296067536401095446991868667192490627309}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{20} + \frac{1706946165484823861549821885493626256594500151380290418124068891540153139582309961}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{19} + \frac{968764866377491321171914041472331338038863400535723139247609407991651600015029274}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{18} + \frac{94904182154752676610246499259773041607409090024036143933492425556935433439733824}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{17} - \frac{631271800486290639599740506382961344102878603548968165308625599826655404901055139}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{16} - \frac{2513532034995219478987939127546983439945988380630759660899290854977573425504413589}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{15} - \frac{276469076603477970372528483818016569106802296861624700247824332206368608936109839}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{14} + \frac{2159192322789045991959741974890183664461368836598656525756898301928854100977837289}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{13} - \frac{433668873759017221158839982340958383888300341220378333688588955995011905026311093}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{12} + \frac{2952631146863172734848862958203608970831557808543805740076810652776602879246083777}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{11} + \frac{3433724273742735993664285199023541154408353341582555365729431335711288764315610972}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{10} + \frac{326779247224522629275212069983569417127127684999510623571353419886521466510202997}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{9} - \frac{459733568488948323798321806084335912189154299604720292233400490055816055182108523}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{8} - \frac{136583061508110217143441424332455567595818321733720613569379139613321325644214289}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{7} - \frac{2434253164212944213009926565151600059508369793126041125737999258164548385623220527}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{6} - \frac{2814184310595598521983035922521487568770700890038330870958976860113747759172250483}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{5} + \frac{1588678641931619982426336519998544359855200298005312542998251551204567210617095504}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{4} + \frac{1514873514991757744676453491818092512460106737191933605248323024013799974992673605}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{3} + \frac{2707627966071299437447166696965419438980735279604677322668674629727230006266888810}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a^{2} + \frac{1513730754733681918016755353996044519477314423548204538268815742056756388161025946}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323} a + \frac{1918443522894778315249219056798607613944495655946197107289899073107810164528680118}{6995867610808007859631606836886493652730041251114999798594182823692306173762483323}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $34$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 24168639235544094000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{35}\cdot(2\pi)^{0}\cdot 24168639235544094000000000000 \cdot 1}{2\sqrt{10607266494966666158512469468409065269845427817856998775062284008873509080731241}}\approx 0.127488294579921$ (assuming GRH)

Galois group

$C_{35}$ (as 35T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 35
The 35 conjugacy class representatives for $C_{35}$
Character table for $C_{35}$ is not computed

Intermediate fields

5.5.1982119441.1, 7.7.88245939632761.1

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 $35$ $35$ $35$ $35$ $35$ $35$ $35$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{7}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{7}$ $35$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{5}$ $35$ $35$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{5}$ $35$ $35$ $35$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
211Data not computed