Properties

Label 35.35.705...625.1
Degree $35$
Signature $[35, 0]$
Discriminant $7.050\times 10^{89}$
Root discriminant $369.06$
Ramified primes $5, 7$
Class number not computed
Class group not computed
Galois group $C_{35}$ (as 35T1)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349)
 
gp: K = bnfinit(x^35 - 175*x^33 - 70*x^32 + 13125*x^31 + 9702*x^30 - 556640*x^29 - 573225*x^28 + 14844900*x^27 + 19050185*x^26 - 261854635*x^25 - 394847250*x^24 + 3128435905*x^23 + 5347852195*x^22 - 25515383355*x^21 - 48297678121*x^20 + 141795944465*x^19 + 292585456625*x^18 - 533394487115*x^17 - 1187150912745*x^16 + 1349113537702*x^15 + 3207931617015*x^14 - 2286376450785*x^13 - 5710072444965*x^12 + 2606036871270*x^11 + 6560227221870*x^10 - 2026242763720*x^9 - 4691478893710*x^8 + 1084067118185*x^7 + 1962115542205*x^6 - 373976828869*x^5 - 432767079290*x^4 + 65717425060*x^3 + 43746645075*x^2 - 3865111915*x - 1645272349, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1645272349, -3865111915, 43746645075, 65717425060, -432767079290, -373976828869, 1962115542205, 1084067118185, -4691478893710, -2026242763720, 6560227221870, 2606036871270, -5710072444965, -2286376450785, 3207931617015, 1349113537702, -1187150912745, -533394487115, 292585456625, 141795944465, -48297678121, -25515383355, 5347852195, 3128435905, -394847250, -261854635, 19050185, 14844900, -573225, -556640, 9702, 13125, -70, -175, 0, 1]);
 

\( x^{35} - 175 x^{33} - 70 x^{32} + 13125 x^{31} + 9702 x^{30} - 556640 x^{29} - 573225 x^{28} + 14844900 x^{27} + 19050185 x^{26} - 261854635 x^{25} - 394847250 x^{24} + 3128435905 x^{23} + 5347852195 x^{22} - 25515383355 x^{21} - 48297678121 x^{20} + 141795944465 x^{19} + 292585456625 x^{18} - 533394487115 x^{17} - 1187150912745 x^{16} + 1349113537702 x^{15} + 3207931617015 x^{14} - 2286376450785 x^{13} - 5710072444965 x^{12} + 2606036871270 x^{11} + 6560227221870 x^{10} - 2026242763720 x^{9} - 4691478893710 x^{8} + 1084067118185 x^{7} + 1962115542205 x^{6} - 373976828869 x^{5} - 432767079290 x^{4} + 65717425060 x^{3} + 43746645075 x^{2} - 3865111915 x - 1645272349 \)

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:  \(705\!\cdots\!625\)\(\medspace = 5^{56}\cdot 7^{60}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $369.06$
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:  $5, 7$
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:  \(1225=5^{2}\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{1225}(1,·)$, $\chi_{1225}(386,·)$, $\chi_{1225}(771,·)$, $\chi_{1225}(1156,·)$, $\chi_{1225}(141,·)$, $\chi_{1225}(526,·)$, $\chi_{1225}(911,·)$, $\chi_{1225}(281,·)$, $\chi_{1225}(666,·)$, $\chi_{1225}(1051,·)$, $\chi_{1225}(36,·)$, $\chi_{1225}(421,·)$, $\chi_{1225}(806,·)$, $\chi_{1225}(1191,·)$, $\chi_{1225}(176,·)$, $\chi_{1225}(561,·)$, $\chi_{1225}(946,·)$, $\chi_{1225}(316,·)$, $\chi_{1225}(701,·)$, $\chi_{1225}(1086,·)$, $\chi_{1225}(71,·)$, $\chi_{1225}(456,·)$, $\chi_{1225}(841,·)$, $\chi_{1225}(211,·)$, $\chi_{1225}(596,·)$, $\chi_{1225}(981,·)$, $\chi_{1225}(351,·)$, $\chi_{1225}(736,·)$, $\chi_{1225}(1121,·)$, $\chi_{1225}(106,·)$, $\chi_{1225}(491,·)$, $\chi_{1225}(876,·)$, $\chi_{1225}(246,·)$, $\chi_{1225}(631,·)$, $\chi_{1225}(1016,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{49} a^{28} + \frac{3}{7} a^{27} - \frac{1}{7} a^{25} - \frac{3}{7} a^{24} + \frac{3}{7} a^{22} + \frac{1}{7} a^{21} + \frac{2}{7} a^{19} - \frac{3}{7} a^{18} - \frac{2}{7} a^{16} - \frac{3}{7} a^{15} - \frac{3}{7} a^{13} - \frac{3}{7} a^{12} + \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{49} a^{7} + \frac{1}{7} a^{6} - \frac{1}{7}$, $\frac{1}{49} a^{29} - \frac{1}{7} a^{26} - \frac{3}{7} a^{25} + \frac{3}{7} a^{23} + \frac{1}{7} a^{22} + \frac{2}{7} a^{20} - \frac{3}{7} a^{19} - \frac{2}{7} a^{17} - \frac{3}{7} a^{16} - \frac{3}{7} a^{14} - \frac{3}{7} a^{13} + \frac{1}{7} a^{11} - \frac{3}{7} a^{10} - \frac{1}{49} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{202027} a^{30} + \frac{17}{202027} a^{29} - \frac{934}{202027} a^{28} - \frac{185}{28861} a^{27} - \frac{4535}{28861} a^{26} + \frac{3018}{28861} a^{25} + \frac{1244}{28861} a^{24} + \frac{13387}{28861} a^{23} + \frac{5391}{28861} a^{22} - \frac{4684}{28861} a^{21} + \frac{4777}{28861} a^{20} - \frac{4803}{28861} a^{19} + \frac{537}{4123} a^{18} - \frac{5511}{28861} a^{17} - \frac{13233}{28861} a^{16} + \frac{459}{1519} a^{15} - \frac{414}{1519} a^{14} - \frac{191}{4123} a^{13} - \frac{12793}{28861} a^{12} + \frac{1294}{4123} a^{11} + \frac{223}{1519} a^{10} - \frac{1851}{10633} a^{9} + \frac{2546}{10633} a^{8} - \frac{52441}{202027} a^{7} - \frac{6394}{28861} a^{6} - \frac{905}{4123} a^{5} - \frac{142}{4123} a^{4} + \frac{20}{217} a^{3} + \frac{3030}{28861} a^{2} - \frac{12757}{28861} a + \frac{9915}{28861}$, $\frac{1}{202027} a^{31} - \frac{1223}{202027} a^{29} - \frac{1909}{202027} a^{28} + \frac{6856}{28861} a^{27} - \frac{6470}{28861} a^{26} - \frac{4709}{28861} a^{25} + \frac{12854}{28861} a^{24} + \frac{8700}{28861} a^{23} - \frac{1502}{28861} a^{22} + \frac{10191}{28861} a^{21} + \frac{571}{28861} a^{20} - \frac{5296}{28861} a^{19} + \frac{8923}{28861} a^{18} - \frac{6129}{28861} a^{17} + \frac{6917}{28861} a^{16} + \frac{463}{1519} a^{15} - \frac{11920}{28861} a^{14} + \frac{1690}{28861} a^{13} - \frac{12595}{28861} a^{12} - \frac{5444}{28861} a^{11} - \frac{2565}{10633} a^{10} - \frac{132}{1519} a^{9} - \frac{66691}{202027} a^{8} + \frac{55123}{202027} a^{7} - \frac{712}{28861} a^{6} - \frac{1249}{4123} a^{5} - \frac{1329}{4123} a^{4} - \frac{13329}{28861} a^{3} - \frac{935}{4123} a^{2} - \frac{216}{1519} a - \frac{7758}{28861}$, $\frac{1}{202027} a^{32} - \frac{1733}{202027} a^{29} - \frac{1695}{202027} a^{28} - \frac{14206}{28861} a^{27} + \frac{1559}{4123} a^{26} - \frac{1565}{4123} a^{25} + \frac{12848}{28861} a^{24} + \frac{131}{1519} a^{23} - \frac{9908}{28861} a^{22} + \frac{11255}{28861} a^{21} - \frac{5316}{28861} a^{20} - \frac{214}{589} a^{19} - \frac{14263}{28861} a^{18} + \frac{3946}{28861} a^{17} - \frac{633}{28861} a^{16} - \frac{398}{931} a^{15} - \frac{3592}{28861} a^{14} + \frac{13823}{28861} a^{13} + \frac{3748}{28861} a^{12} - \frac{23843}{202027} a^{11} + \frac{696}{1519} a^{10} + \frac{5708}{28861} a^{9} + \frac{43229}{202027} a^{8} + \frac{50660}{202027} a^{7} - \frac{11397}{28861} a^{6} + \frac{943}{4123} a^{5} + \frac{12032}{28861} a^{4} + \frac{2029}{4123} a^{3} + \frac{34}{133} a^{2} - \frac{4014}{28861} a + \frac{8548}{28861}$, $\frac{1}{122630389} a^{33} - \frac{4}{122630389} a^{32} - \frac{281}{122630389} a^{31} - \frac{9}{3955819} a^{30} - \frac{103889}{17518627} a^{29} + \frac{1200830}{122630389} a^{28} - \frac{8308166}{17518627} a^{27} - \frac{9245}{18817} a^{26} + \frac{750378}{17518627} a^{25} + \frac{449578}{922033} a^{24} + \frac{2268655}{17518627} a^{23} + \frac{2444434}{17518627} a^{22} - \frac{2587630}{17518627} a^{21} - \frac{6372777}{17518627} a^{20} - \frac{1340582}{17518627} a^{19} - \frac{93644}{17518627} a^{18} - \frac{3210789}{17518627} a^{17} + \frac{1024582}{17518627} a^{16} + \frac{243493}{17518627} a^{15} + \frac{1967683}{17518627} a^{14} + \frac{4790171}{17518627} a^{13} + \frac{58513209}{122630389} a^{12} + \frac{16188141}{122630389} a^{11} - \frac{17309172}{122630389} a^{10} - \frac{55381439}{122630389} a^{9} - \frac{231898}{2502661} a^{8} + \frac{54440133}{122630389} a^{7} + \frac{5971351}{17518627} a^{6} + \frac{821764}{17518627} a^{5} + \frac{2260521}{17518627} a^{4} + \frac{1358540}{17518627} a^{3} + \frac{92707}{17518627} a^{2} + \frac{2000}{11533} a + \frac{8068836}{17518627}$, $\frac{1}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{34} + \frac{17377579808658439738813663136930653007035750669640973924662982408789101166003874760535546808172166375151473078220109514907070618293409860186747815296229907652704707}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{33} - \frac{7449836994492372784940877931449948653089464892018866255397836239014542485915418223983770376351165700566273311582346926438202439248428315717664732518522126115290150882}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{32} + \frac{4755867393373538561106285858174561489350908587406203338030286492097178464408427359119582149879671997092128246950231072988690587806095732747035088762773557725881124517}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{31} - \frac{11631952682191213973687994737769536633028568080186179344202242778205685933595331677806794621016413853580725340115500713013694958835065290654586063038584536524463642633}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{30} - \frac{24150082474374701898022074399716589910374996586129699424981643494648428565248795791097406004504584874057447570482883320869540536848049433468021868066773078038239794166441}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{29} + \frac{495429178248738986621475723915805222762477372692057907771660337899947842824713185428217912154350387483046452412401240275069334093570322942763333237832651858181823177855}{567977411198584004278006868143082265340571944969273413438852899952387175647894115132577819596988636429341075710038217399927850957056643443783883980616134578464836624248669} a^{28} - \frac{716433833429610684161079039532727689395054277074203675824157924203799940866429513615927771929678619379938526947087342366981236168359819045433397420972140231928406110107595}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{27} - \frac{1067739291986139121772714104760241140988720561179591191353030201933120920712939912733098836220118013905065838508781461485376920135677016482734925802230652289222359717008}{220236139036185634311880214177929857989201366416657037864045002022354210965509963010591399435567022288928172214096451644869983024164820927181506033300133816139426446137239} a^{26} - \frac{309750367858372489430740641204282579337358426815947282759326887535288142291355650656043616579933198398881281634657098002865262890918485644467531996218837289901337619556683}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{25} + \frac{138146563464342498163987906532335148321723236099393850398293285350288827594380195079960857055074180683260298490113477915942718342329556066108612947123751526554151266474712}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{24} - \frac{357207665900610661607085257151986413694316932908787820719834590359202798116790621992715776518380931095499877319933139037816055351947804372724505389556014241429677265667010}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{23} - \frac{770031487312612151919245381703340140200917929822396927379794780644295336460000478879356601003858540338057652921301491981507772243284248442626488250268260896691365976025166}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{22} + \frac{676443961793319457079011139847094109676486630995730632094400555045222629186813220712372948604472123350602564184620982039855164534628648557728011794374570144743291451507394}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{21} - \frac{586074898683743065157582914963019781279389601643648645192279019332564076177979671109738596417534335493920300545610537501278720723128322916005853715233764786301957409286733}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{20} + \frac{162037171574880283538248607646673429410210512698834608646260619294863301693796414148324099702634503944905313743820471554064015184948323727901432773332169372747831271264885}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{19} + \frac{170335061280542222836585836719336582551438519442471968367158361809058277413496660166789132930033000005171072572897727743420936436426125846478737239016635033066499545754974}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{18} - \frac{407913129761903688201683337104022240742460939498412317225816880821065451379862835531355374432901174674013211044648013668183770546969886485136457491868531192405778009270253}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{17} + \frac{708913532412490185992489119589528501438798292343560327112930755372849704971158967038513999951732538192423172108239880489520499995265449249045005461285598187714044193208274}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{16} - \frac{754073067921616709299741386820804643386250233724780144330487062959635210591552467885644783322428042849259262731930387767507769379205167277734323759282839048122045233299719}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{15} + \frac{713353090837133083402589550047904954348390327722740912769173756209813582126055090698129278836831402313993772441596164406831050703202285543831208917967948454620234399464813}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{14} + \frac{3612751566058515122472384909140504622833353678465347156748146818673624093827915402832057626606933443651850730328194700073284409499903398529943846325640449678736428855728734}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{13} + \frac{95529464469508484484447079780725242305921784719765324650370785756887812352904231635088768912055332370048279698729084719176155052433860232535411370517107016087230005644192}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{12} - \frac{5028443898021525170705667702230768642707723674076309964944501061858105742996731342670350688795149446680567006749458089039109540523118736518519161575670569790162385139331820}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{11} + \frac{2045665466058137576480250213404517209183057527056038561814748865145526370564087155848226201157410216766441237841453252834776133902704640830025439642151180649285816329679870}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{10} - \frac{111366518110135839390252846594332801945221382407395006954466531034257478064754247083756397279286727211704630069439857223491167618097791459290456923859715130527689306941457}{567977411198584004278006868143082265340571944969273413438852899952387175647894115132577819596988636429341075710038217399927850957056643443783883980616134578464836624248669} a^{9} - \frac{1794839372522755423679013190421672741340706397832202590616891515493708295496166948534993245044562725649117186908345685930044798492049502633257246196950755082797831512727686}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{8} + \frac{4689919442872212452956379131406414225507753297709561476356067199586855974178719355174526925405808568437772893975092538613557061116468164959363753400454394781386907050304342}{10791570812773096081282130494718563041470866954416194855338205099095356337309988187518978572342784092157480438490726130598629168184076225431893795631706556990831895860724711} a^{7} + \frac{246077310423257461236227057170941348646590156332474559935865223305404321083100785807148423535219664658067750728874544825246249486145104031971523483661437505270826718196137}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{6} - \frac{70085852759735076573071579435942654177428300893495904745157038596695836506074528948306628474630844510120129928039485218446788924995891084986563479610576165943618074321703}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{5} + \frac{364058652331192180128736932641002313223593696603189219648729823157809459937615834720888410924214803606004787949531868954850605291521645383755319174847196657724223630733376}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{4} + \frac{11861927833357066424366957910728755558101576659586860561534270053414156680502230401226057306910014062354147012964351016278520130292717995522451565070721285729283707578517}{81139630171226286325429552591868895048653134995610487634121842850341025092556302161796831370998376632763010815719745342846835851008091920540554854373733511209262374892667} a^{3} - \frac{25735528504567377917450214297133401288247334270156340764478872420227558070036425691188568342263911571747597802508180471900183789532159459855118664586674631292242155427052}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a^{2} + \frac{431998744325000940144674790989098599656061784275352257077306614473297436392733427607757297831158921098297457556234733762213183993024042432916897562014055547138765949440216}{1541652973253299440183161499245509005924409564916599265048315014156479476758569741074139796048969156022497205498675161514089881169153746490270542233100936712975985122960673} a + \frac{1421994630833843036941009192290305636425013960468156153634898910156401752705616715898450040479326245133526553442184310774679871802269259089077573030155568731098464949285}{49730741072687078715585854814371258255626114997309653711235968198596112153502249712069025678999005032983780822537908435938383263521088596460340072035514087515354358805183}$

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

Class group and class number

not computed

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:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) $ not computed

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

5.5.390625.1, 7.7.13841287201.1

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 $35$ $35$ R R $35$ $35$ $35$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{7}$ $35$ $35$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{7}$ $35$ $35$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{5}$ $35$ $35$ $35$

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
5Data not computed
$7$7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$
7.7.12.1$x^{7} - 7 x^{6} + 7$$7$$1$$12$$C_7$$[2]$