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Properties

Label 35.35.107...681.1
Degree $35$
Signature $[35, 0]$
Discriminant $1.074\times 10^{73}$
Root discriminant $122.07$
Ramified primes $11, 29$
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 - 2*x^34 - 91*x^33 + 186*x^32 + 3529*x^31 - 7282*x^30 - 77406*x^29 + 160015*x^28 + 1074054*x^27 - 2215841*x^26 - 9975621*x^25 + 20528466*x^24 + 63908089*x^23 - 131541229*x^22 - 286306729*x^21 + 593348019*x^20 + 897545333*x^19 - 1896502635*x^18 - 1942677455*x^17 + 4283320775*x^16 + 2804705170*x^15 - 6752081061*x^14 - 2493142459*x^13 + 7254188151*x^12 + 1058706744*x^11 - 5105442290*x^10 + 147588576*x^9 + 2206890342*x^8 - 365733379*x^7 - 523140235*x^6 + 145322301*x^5 + 51716162*x^4 - 20476030*x^3 - 56585*x^2 + 426253*x - 7523)
 
gp: K = bnfinit(x^35 - 2*x^34 - 91*x^33 + 186*x^32 + 3529*x^31 - 7282*x^30 - 77406*x^29 + 160015*x^28 + 1074054*x^27 - 2215841*x^26 - 9975621*x^25 + 20528466*x^24 + 63908089*x^23 - 131541229*x^22 - 286306729*x^21 + 593348019*x^20 + 897545333*x^19 - 1896502635*x^18 - 1942677455*x^17 + 4283320775*x^16 + 2804705170*x^15 - 6752081061*x^14 - 2493142459*x^13 + 7254188151*x^12 + 1058706744*x^11 - 5105442290*x^10 + 147588576*x^9 + 2206890342*x^8 - 365733379*x^7 - 523140235*x^6 + 145322301*x^5 + 51716162*x^4 - 20476030*x^3 - 56585*x^2 + 426253*x - 7523, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7523, 426253, -56585, -20476030, 51716162, 145322301, -523140235, -365733379, 2206890342, 147588576, -5105442290, 1058706744, 7254188151, -2493142459, -6752081061, 2804705170, 4283320775, -1942677455, -1896502635, 897545333, 593348019, -286306729, -131541229, 63908089, 20528466, -9975621, -2215841, 1074054, 160015, -77406, -7282, 3529, 186, -91, -2, 1]);
 

\( x^{35} - 2 x^{34} - 91 x^{33} + 186 x^{32} + 3529 x^{31} - 7282 x^{30} - 77406 x^{29} + 160015 x^{28} + 1074054 x^{27} - 2215841 x^{26} - 9975621 x^{25} + 20528466 x^{24} + 63908089 x^{23} - 131541229 x^{22} - 286306729 x^{21} + 593348019 x^{20} + 897545333 x^{19} - 1896502635 x^{18} - 1942677455 x^{17} + 4283320775 x^{16} + 2804705170 x^{15} - 6752081061 x^{14} - 2493142459 x^{13} + 7254188151 x^{12} + 1058706744 x^{11} - 5105442290 x^{10} + 147588576 x^{9} + 2206890342 x^{8} - 365733379 x^{7} - 523140235 x^{6} + 145322301 x^{5} + 51716162 x^{4} - 20476030 x^{3} - 56585 x^{2} + 426253 x - 7523 \)

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:  \(107\!\cdots\!681\)\(\medspace = 11^{28}\cdot 29^{30}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $122.07$
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:  $11, 29$
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:  \(319=11\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{319}(256,·)$, $\chi_{319}(1,·)$, $\chi_{319}(262,·)$, $\chi_{319}(257,·)$, $\chi_{319}(136,·)$, $\chi_{319}(268,·)$, $\chi_{319}(141,·)$, $\chi_{319}(16,·)$, $\chi_{319}(146,·)$, $\chi_{319}(20,·)$, $\chi_{319}(23,·)$, $\chi_{319}(152,·)$, $\chi_{319}(25,·)$, $\chi_{319}(284,·)$, $\chi_{319}(291,·)$, $\chi_{319}(36,·)$, $\chi_{319}(168,·)$, $\chi_{319}(169,·)$, $\chi_{319}(170,·)$, $\chi_{319}(45,·)$, $\chi_{319}(49,·)$, $\chi_{319}(306,·)$, $\chi_{319}(53,·)$, $\chi_{319}(313,·)$, $\chi_{319}(59,·)$, $\chi_{319}(190,·)$, $\chi_{319}(181,·)$, $\chi_{319}(199,·)$, $\chi_{319}(78,·)$, $\chi_{319}(81,·)$, $\chi_{319}(82,·)$, $\chi_{319}(223,·)$, $\chi_{319}(103,·)$, $\chi_{319}(210,·)$, $\chi_{319}(111,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $\frac{1}{17} a^{30} + \frac{1}{17} a^{29} + \frac{8}{17} a^{28} + \frac{4}{17} a^{27} + \frac{6}{17} a^{26} - \frac{7}{17} a^{25} + \frac{4}{17} a^{24} - \frac{1}{17} a^{23} + \frac{6}{17} a^{22} - \frac{3}{17} a^{21} + \frac{2}{17} a^{20} + \frac{1}{17} a^{19} + \frac{4}{17} a^{18} + \frac{2}{17} a^{17} - \frac{2}{17} a^{16} - \frac{5}{17} a^{15} - \frac{1}{17} a^{14} - \frac{1}{17} a^{13} - \frac{3}{17} a^{12} + \frac{7}{17} a^{11} - \frac{6}{17} a^{10} + \frac{8}{17} a^{9} - \frac{5}{17} a^{8} + \frac{3}{17} a^{7} - \frac{1}{17} a^{6} + \frac{8}{17} a^{5} + \frac{5}{17} a^{4} - \frac{2}{17} a^{3} + \frac{4}{17} a^{2} + \frac{2}{17}$, $\frac{1}{17} a^{31} + \frac{7}{17} a^{29} - \frac{4}{17} a^{28} + \frac{2}{17} a^{27} + \frac{4}{17} a^{26} - \frac{6}{17} a^{25} - \frac{5}{17} a^{24} + \frac{7}{17} a^{23} + \frac{8}{17} a^{22} + \frac{5}{17} a^{21} - \frac{1}{17} a^{20} + \frac{3}{17} a^{19} - \frac{2}{17} a^{18} - \frac{4}{17} a^{17} - \frac{3}{17} a^{16} + \frac{4}{17} a^{15} - \frac{2}{17} a^{13} - \frac{7}{17} a^{12} + \frac{4}{17} a^{11} - \frac{3}{17} a^{10} + \frac{4}{17} a^{9} + \frac{8}{17} a^{8} - \frac{4}{17} a^{7} - \frac{8}{17} a^{6} - \frac{3}{17} a^{5} - \frac{7}{17} a^{4} + \frac{6}{17} a^{3} - \frac{4}{17} a^{2} + \frac{2}{17} a - \frac{2}{17}$, $\frac{1}{16442519} a^{32} + \frac{33176}{16442519} a^{31} - \frac{63573}{16442519} a^{30} - \frac{3342209}{16442519} a^{29} + \frac{20668}{967207} a^{28} + \frac{5025103}{16442519} a^{27} + \frac{2217578}{16442519} a^{26} - \frac{3198150}{16442519} a^{25} + \frac{6334043}{16442519} a^{24} + \frac{6328100}{16442519} a^{23} + \frac{4922903}{16442519} a^{22} + \frac{160167}{16442519} a^{21} - \frac{2229301}{16442519} a^{20} - \frac{66887}{16442519} a^{19} - \frac{5000589}{16442519} a^{18} + \frac{3165072}{16442519} a^{17} + \frac{962738}{16442519} a^{16} + \frac{7182043}{16442519} a^{15} - \frac{6338095}{16442519} a^{14} - \frac{4972899}{16442519} a^{13} + \frac{6556637}{16442519} a^{12} + \frac{6554077}{16442519} a^{11} + \frac{2978377}{16442519} a^{10} - \frac{1963915}{16442519} a^{9} + \frac{453865}{967207} a^{8} + \frac{6278991}{16442519} a^{7} + \frac{3003927}{16442519} a^{6} - \frac{124675}{967207} a^{5} + \frac{6635400}{16442519} a^{4} + \frac{4110072}{16442519} a^{3} + \frac{191029}{967207} a^{2} + \frac{5576237}{16442519} a - \frac{3740817}{16442519}$, $\frac{1}{5047853333} a^{33} + \frac{45}{5047853333} a^{32} + \frac{86578153}{5047853333} a^{31} + \frac{75628982}{5047853333} a^{30} - \frac{1283099349}{5047853333} a^{29} - \frac{599918663}{5047853333} a^{28} + \frac{1197306054}{5047853333} a^{27} - \frac{1657687911}{5047853333} a^{26} + \frac{77259500}{296932549} a^{25} + \frac{1229634698}{5047853333} a^{24} + \frac{1817813076}{5047853333} a^{23} - \frac{1928066267}{5047853333} a^{22} - \frac{828619354}{5047853333} a^{21} - \frac{12497288}{5047853333} a^{20} - \frac{846308719}{5047853333} a^{19} - \frac{2453847993}{5047853333} a^{18} - \frac{1586543062}{5047853333} a^{17} - \frac{2486197835}{5047853333} a^{16} - \frac{15879486}{5047853333} a^{15} + \frac{2351158649}{5047853333} a^{14} + \frac{1786392285}{5047853333} a^{13} + \frac{1103348119}{5047853333} a^{12} + \frac{1387701249}{5047853333} a^{11} + \frac{1874201832}{5047853333} a^{10} - \frac{250010003}{5047853333} a^{9} - \frac{1907904745}{5047853333} a^{8} - \frac{1201603635}{5047853333} a^{7} + \frac{566191483}{5047853333} a^{6} - \frac{2016630826}{5047853333} a^{5} - \frac{511402422}{5047853333} a^{4} + \frac{2464165785}{5047853333} a^{3} - \frac{1263983250}{5047853333} a^{2} + \frac{1185824816}{5047853333} a - \frac{349091473}{5047853333}$, $\frac{1}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{34} - \frac{576587184938242850960602237972132869415751087158990658229244846632560553403859558173048721862419073}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{33} + \frac{102251208161142123469008521108580096895685925487945475523318629467882728503271194587845063829031794895}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{32} - \frac{255258091232409541042324389887770766865775474179374300455866430626757157983305966744727042766529585738054478}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{31} - \frac{11474089667414270294641164064831757762180802649529391909109693971098475654096084249651855141483742370262404}{686787522071089965709245337005070489073162227158677713648698703247835231702051974502932529655104196694097359} a^{30} + \frac{2303312124217923611587042728864315954622128718253360607397526740115178330091288518716780297352345768145681803}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{29} + \frac{191700209918350792600671535711772402892365204620650561818895118462762845203734465437319875757639703584954575}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{28} + \frac{5284341934614398762758768323787711192429458266136581462485323585018989252681463774616250162419059012579302005}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{27} - \frac{3815484871391286332585844082632516545643650179164007281495640564279713685452576327895545295700391766868427742}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{26} - \frac{3117778762965619425303004852805587923955018281952919583751919868372867235257032881524332269956634611202747484}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{25} - \frac{4319522039270293314350292344645687800540024023030410258718665619408145252134022209512585539563787605563370335}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{24} - \frac{35267545039419564721710545143897689059212043033501924894354422826523390167518060320399573366129037173993607}{686787522071089965709245337005070489073162227158677713648698703247835231702051974502932529655104196694097359} a^{23} + \frac{989498021090140654667894055235692971946721755349844947822218734632525800580089281203845848296189063178476142}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{22} - \frac{1380923208333966687836644528130726154062180688256056262133727903071427036040953821829872814092050967561330361}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{21} - \frac{1835001445151220513030032156094378478429217372763994960007793704786943533127848950697771546690153393745551716}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{20} - \frac{4309492981051535097088381577787518356055916868913522646160904727037907727188805593023851722172644638339636253}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{19} + \frac{4300189011579257439239662477657053553614211157496699958212204669384137116677370248910590960208061580050807877}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{18} + \frac{4814315233009148800124602501681751155068020001367949025776876401399907083995078498949785086108735502597030315}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{17} + \frac{1173903747199473745757749013904970139598266607039434534530067241656685031136185875805047752659971903911506410}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{16} - \frac{3860001771660375862597421813113854905616154160492938871722246151613075491926617719273939858675403040032279368}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{15} + \frac{4957704585061834997351688183885364512160095188516339438562210270365072111741072704452777182261461782366257312}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{14} - \frac{5334488806344844306317296737745705805662234140880654507878219143032863109184542948126263491406546227387445550}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{13} + \frac{1767308912535067576754518858181355865925210739699929598685534290062332183235403199492771650658907076037406703}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{12} + \frac{503547065800080245411870134946989516635231694299419612983633897055157576111272259208742263818307391065190286}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{11} - \frac{4544845528546366141386332424843512319947804973565892620178618122278730037478072553841025551859742592172861643}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{10} + \frac{681349962243059575430472874345256534685965613693211119275570447005610894218669752327318462067616381850066836}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{9} - \frac{3596617282619364067323659734974543126271112373587864424586377983665666163751795128799844383241978880299328777}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{8} - \frac{4436318164569792113824747624845807363596100820697318201574301223913903171467857174018193267358528206625525387}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{7} + \frac{3062275136439487443086624597907433842051197594157663485494798709374488495522981916629898224983816929021650565}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{6} - \frac{5819776375558770810462382245239297986032289335433376265631536430199102687043769847324277078736018041899975925}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{5} - \frac{5345406706248911181594867938875624411558167999655043137565780580070774603398448205527943988146862302870421208}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{4} - \frac{259252459344044489022089702232691453699379355427226418573593301159833269916658198092109650023661479529862769}{686787522071089965709245337005070489073162227158677713648698703247835231702051974502932529655104196694097359} a^{3} - \frac{5231533075857175214061540036084911251326417886416547259301855927016594173512790793682292375400092777664242621}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a^{2} + \frac{1417746385162257870917441472744358427191353520289805341123159869392000597593727280592170834935641417844470086}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103} a - \frac{4204738957663277489616643690482094611731200436592049788091188638038662554129678791722387433339185102808422129}{11675387875208529417057170729086198314243757861697521132027877955213198938934883566549853004136771343799655103}$

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:  \( 21825601175888975000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{35}\cdot(2\pi)^{0}\cdot 21825601175888975000000000 \cdot 1}{2\sqrt{10738289826578710854795473611086878947834443845310364463749065797704132681}}\approx 0.114424363912813$ (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

\(\Q(\zeta_{11})^+\), 7.7.594823321.1

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 $35$ $35$ $35$ $35$ R $35$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{7}$ $35$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{5}$ R $35$ $35$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{5}$ $35$ $35$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{7}$

In the table, R denotes a ramified prime. 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
11Data not computed
29Data not computed