# Properties

 Label 42.42.5239206534...0301.1 Degree $42$ Signature $[42, 0]$ Discriminant $7^{21}\cdot 43^{41}$ Root discriminant $104.02$ Ramified primes $7, 43$ Class number Not computed Class group Not computed Galois group $C_{42}$ (as 42T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - x^41 - 85*x^40 + 85*x^39 + 3355*x^38 - 3355*x^37 - 81613*x^36 + 81613*x^35 + 1369379*x^34 - 1369379*x^33 - 16806205*x^32 + 16806205*x^31 + 156107459*x^30 - 156107459*x^29 - 1120160061*x^28 + 1120160061*x^27 + 6282191555*x^26 - 6282191555*x^25 - 27681539389*x^24 + 27681539389*x^23 + 95822936771*x^22 - 95822936771*x^21 - 259252432189*x^20 + 259252432189*x^19 + 542530659011*x^18 - 542530659011*x^17 - 863673531709*x^16 + 863673531709*x^15 + 1020501541571*x^14 - 1020501541571*x^13 - 863673531709*x^12 + 863673531709*x^11 + 497119576771*x^10 - 497119576771*x^9 - 180198260029*x^8 + 180198260029*x^7 + 36543447747*x^6 - 36543447747*x^5 - 3382656317*x^4 + 3382656317*x^3 + 89178819*x^2 - 89178819*x - 998717)

gp: K = bnfinit(x^42 - x^41 - 85*x^40 + 85*x^39 + 3355*x^38 - 3355*x^37 - 81613*x^36 + 81613*x^35 + 1369379*x^34 - 1369379*x^33 - 16806205*x^32 + 16806205*x^31 + 156107459*x^30 - 156107459*x^29 - 1120160061*x^28 + 1120160061*x^27 + 6282191555*x^26 - 6282191555*x^25 - 27681539389*x^24 + 27681539389*x^23 + 95822936771*x^22 - 95822936771*x^21 - 259252432189*x^20 + 259252432189*x^19 + 542530659011*x^18 - 542530659011*x^17 - 863673531709*x^16 + 863673531709*x^15 + 1020501541571*x^14 - 1020501541571*x^13 - 863673531709*x^12 + 863673531709*x^11 + 497119576771*x^10 - 497119576771*x^9 - 180198260029*x^8 + 180198260029*x^7 + 36543447747*x^6 - 36543447747*x^5 - 3382656317*x^4 + 3382656317*x^3 + 89178819*x^2 - 89178819*x - 998717, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-998717, -89178819, 89178819, 3382656317, -3382656317, -36543447747, 36543447747, 180198260029, -180198260029, -497119576771, 497119576771, 863673531709, -863673531709, -1020501541571, 1020501541571, 863673531709, -863673531709, -542530659011, 542530659011, 259252432189, -259252432189, -95822936771, 95822936771, 27681539389, -27681539389, -6282191555, 6282191555, 1120160061, -1120160061, -156107459, 156107459, 16806205, -16806205, -1369379, 1369379, 81613, -81613, -3355, 3355, 85, -85, -1, 1]);

## Normalizeddefining polynomial

$$x^{42} - x^{41} - 85 x^{40} + 85 x^{39} + 3355 x^{38} - 3355 x^{37} - 81613 x^{36} + 81613 x^{35} + 1369379 x^{34} - 1369379 x^{33} - 16806205 x^{32} + 16806205 x^{31} + 156107459 x^{30} - 156107459 x^{29} - 1120160061 x^{28} + 1120160061 x^{27} + 6282191555 x^{26} - 6282191555 x^{25} - 27681539389 x^{24} + 27681539389 x^{23} + 95822936771 x^{22} - 95822936771 x^{21} - 259252432189 x^{20} + 259252432189 x^{19} + 542530659011 x^{18} - 542530659011 x^{17} - 863673531709 x^{16} + 863673531709 x^{15} + 1020501541571 x^{14} - 1020501541571 x^{13} - 863673531709 x^{12} + 863673531709 x^{11} + 497119576771 x^{10} - 497119576771 x^{9} - 180198260029 x^{8} + 180198260029 x^{7} + 36543447747 x^{6} - 36543447747 x^{5} - 3382656317 x^{4} + 3382656317 x^{3} + 89178819 x^{2} - 89178819 x - 998717$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $42$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[42, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$5239206534209069133889646882729090965733830111939046184442828436267483950448112600301=7^{21}\cdot 43^{41}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $104.02$ 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: $7, 43$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $42$ This field is Galois and abelian over $\Q$. Conductor: $$301=7\cdot 43$$ Dirichlet character group: $\lbrace$$\chi_{301}(1,·), \chi_{301}(132,·), \chi_{301}(265,·), \chi_{301}(267,·), \chi_{301}(15,·), \chi_{301}(274,·), \chi_{301}(20,·), \chi_{301}(281,·), \chi_{301}(27,·), \chi_{301}(286,·), \chi_{301}(69,·), \chi_{301}(48,·), \chi_{301}(34,·), \chi_{301}(36,·), \chi_{301}(169,·), \chi_{301}(300,·), \chi_{301}(174,·), \chi_{301}(125,·), \chi_{301}(176,·), \chi_{301}(55,·), \chi_{301}(57,·), \chi_{301}(62,·), \chi_{301}(64,·), \chi_{301}(197,·), \chi_{301}(202,·), \chi_{301}(183,·), \chi_{301}(76,·), \chi_{301}(78,·), \chi_{301}(209,·), \chi_{301}(92,·), \chi_{301}(118,·), \chi_{301}(223,·), \chi_{301}(225,·), \chi_{301}(99,·), \chi_{301}(104,·), \chi_{301}(237,·), \chi_{301}(239,·), \chi_{301}(232,·), \chi_{301}(244,·), \chi_{301}(246,·), \chi_{301}(253,·), \chi_{301}(127,·)$$\rbrace$ This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{2209943} a^{22} - \frac{854722}{2209943} a^{21} - \frac{44}{2209943} a^{20} + \frac{539236}{2209943} a^{19} + \frac{836}{2209943} a^{18} - \frac{866476}{2209943} a^{17} - \frac{8976}{2209943} a^{16} - \frac{929326}{2209943} a^{15} + \frac{59840}{2209943} a^{14} - \frac{629881}{2209943} a^{13} - \frac{256256}{2209943} a^{12} - \frac{416390}{2209943} a^{11} + \frac{704704}{2209943} a^{10} + \frac{772293}{2209943} a^{9} + \frac{1001879}{2209943} a^{8} - \frac{1089697}{2209943} a^{7} - \frac{1001879}{2209943} a^{6} - \frac{54836}{2209943} a^{5} - \frac{619520}{2209943} a^{4} - \frac{1043737}{2209943} a^{3} + \frac{123904}{2209943} a^{2} + \frac{154043}{2209943} a - \frac{4096}{2209943}$, $\frac{1}{2209943} a^{23} - \frac{46}{2209943} a^{21} + \frac{500499}{2209943} a^{20} + \frac{920}{2209943} a^{19} - \frac{130473}{2209943} a^{18} - \frac{10488}{2209943} a^{17} + \frac{8098}{2209943} a^{16} + \frac{75072}{2209943} a^{15} - \frac{986193}{2209943} a^{14} - \frac{350336}{2209943} a^{13} - \frac{606492}{2209943} a^{12} + \frac{1071616}{2209943} a^{11} + \frac{190102}{2209943} a^{10} + \frac{104983}{2209943} a^{9} + \frac{539757}{2209943} a^{8} + \frac{316009}{2209943} a^{7} + \frac{525653}{2209943} a^{6} + \frac{525975}{2209943} a^{5} - \frac{604776}{2209943} a^{4} + \frac{518144}{2209943} a^{3} + \frac{950228}{2209943} a^{2} - \frac{47104}{2209943} a - \frac{391600}{2209943}$, $\frac{1}{2209943} a^{24} + \frac{962261}{2209943} a^{21} - \frac{1104}{2209943} a^{20} + \frac{365010}{2209943} a^{19} + \frac{27968}{2209943} a^{18} - \frac{70824}{2209943} a^{17} - \frac{337824}{2209943} a^{16} + \frac{463671}{2209943} a^{15} + \frac{192361}{2209943} a^{14} - \frac{851759}{2209943} a^{13} + \frac{333555}{2209943} a^{12} + \frac{925649}{2209943} a^{11} - \frac{627778}{2209943} a^{10} + \frac{706147}{2209943} a^{9} - \frac{6360}{2209943} a^{8} - \frac{981663}{2209943} a^{7} + \frac{848344}{2209943} a^{6} - \frac{917289}{2209943} a^{5} + \frac{749483}{2209943} a^{4} - \frac{652871}{2209943} a^{3} - \frac{977349}{2209943} a^{2} + \frac{64549}{2209943} a - \frac{188416}{2209943}$, $\frac{1}{2209943} a^{25} - \frac{1200}{2209943} a^{21} + \frac{715577}{2209943} a^{20} + \frac{32000}{2209943} a^{19} - \frac{101768}{2209943} a^{18} - \frac{410400}{2209943} a^{17} - \frac{948780}{2209943} a^{16} + \frac{923497}{2209943} a^{15} - \frac{275191}{2209943} a^{14} + \frac{237601}{2209943} a^{13} + \frac{640525}{2209943} a^{12} - \frac{695546}{2209943} a^{11} - \frac{719705}{2209943} a^{10} - \frac{1068451}{2209943} a^{9} - \frac{115876}{2209943} a^{8} + \frac{18621}{2209943} a^{7} + \frac{426867}{2209943} a^{6} + \frac{484668}{2209943} a^{5} + \frac{527770}{2209943} a^{4} + \frac{266627}{2209943} a^{3} + \frac{712398}{2209943} a^{2} - \frac{42857}{2209943} a + \frac{1092687}{2209943}$, $\frac{1}{2209943} a^{26} + \frac{462729}{2209943} a^{21} - \frac{20800}{2209943} a^{20} - \frac{531867}{2209943} a^{19} + \frac{592800}{2209943} a^{18} + \frac{163173}{2209943} a^{17} - \frac{1007931}{2209943} a^{16} + \frac{554824}{2209943} a^{15} - \frac{882518}{2209943} a^{14} + \frac{583831}{2209943} a^{13} - \frac{1020669}{2209943} a^{12} - \frac{940587}{2209943} a^{11} + \frac{378123}{2209943} a^{10} + \frac{669607}{2209943} a^{9} + \frac{64429}{2209943} a^{8} + \frac{1076723}{2209943} a^{7} + \frac{438860}{2209943} a^{6} + \frac{3980}{8599} a^{5} - \frac{616525}{2209943} a^{4} - \frac{944264}{2209943} a^{3} + \frac{575762}{2209943} a^{2} + \frac{309075}{2209943} a - \frac{495314}{2209943}$, $\frac{1}{2209943} a^{27} - \frac{23400}{2209943} a^{21} - \frac{61278}{2209943} a^{20} + \frac{702000}{2209943} a^{19} + \frac{61754}{2209943} a^{18} - \frac{763588}{2209943} a^{17} - \frac{682512}{2209943} a^{16} - \frac{970405}{2209943} a^{15} - \frac{743682}{2209943} a^{14} + \frac{432139}{2209943} a^{13} - \frac{559571}{2209943} a^{12} + \frac{16035}{2209943} a^{11} - \frac{378187}{2209943} a^{10} + \frac{949533}{2209943} a^{9} + \frac{31586}{2209943} a^{8} + \frac{987435}{2209943} a^{7} - \frac{141946}{2209943} a^{6} - \frac{974607}{2209943} a^{5} - \frac{460258}{2209943} a^{4} + \frac{380986}{2209943} a^{3} + \frac{1096251}{2209943} a^{2} + \frac{1052804}{2209943} a - \frac{793110}{2209943}$, $\frac{1}{2209943} a^{28} - \frac{571928}{2209943} a^{21} - \frac{327600}{2209943} a^{20} - \frac{590376}{2209943} a^{19} - \frac{1090675}{2209943} a^{18} + \frac{6113}{2209943} a^{17} - \frac{1064220}{2209943} a^{16} + \frac{1076981}{2209943} a^{15} - \frac{415723}{2209943} a^{14} + \frac{544839}{2209943} a^{13} - \frac{799006}{2209943} a^{12} - \frac{265500}{2209943} a^{11} + \frac{428467}{2209943} a^{10} + \frac{983875}{2209943} a^{9} - \frac{329252}{2209943} a^{8} - \frac{729412}{2209943} a^{7} + \frac{342080}{2209943} a^{6} + \frac{354225}{2209943} a^{5} + \frac{839066}{2209943} a^{4} - \frac{269456}{2209943} a^{3} + \frac{961188}{2209943} a^{2} - \frac{603943}{2209943} a - \frac{818851}{2209943}$, $\frac{1}{2209943} a^{29} - \frac{380016}{2209943} a^{21} + \frac{764108}{2209943} a^{20} - \frac{1099146}{2209943} a^{19} + \frac{790233}{2209943} a^{18} - \frac{911742}{2209943} a^{17} - \frac{1061101}{2209943} a^{16} + \frac{994793}{2209943} a^{15} - \frac{670882}{2209943} a^{14} - \frac{151258}{2209943} a^{13} + \frac{962749}{2209943} a^{12} + \frac{996170}{2209943} a^{11} + \frac{368619}{2209943} a^{10} + \frac{984071}{2209943} a^{9} - \frac{937512}{2209943} a^{8} + \frac{351637}{2209943} a^{7} + \frac{562325}{2209943} a^{6} - \frac{103629}{2209943} a^{5} - \frac{942826}{2209943} a^{4} - \frac{490360}{2209943} a^{3} - \frac{469269}{2209943} a^{2} - \frac{901585}{2209943} a - \frac{77508}{2209943}$, $\frac{1}{2209943} a^{30} - \frac{899019}{2209943} a^{21} - \frac{140306}{2209943} a^{20} - \frac{76609}{2209943} a^{19} + \frac{759785}{2209943} a^{18} - \frac{927546}{2209943} a^{17} - \frac{86774}{2209943} a^{16} + \frac{521017}{2209943} a^{15} - \frac{307288}{2209943} a^{14} - \frac{549131}{2209943} a^{13} + \frac{754369}{2209943} a^{12} - \frac{364878}{2209943} a^{11} + \frac{1096538}{2209943} a^{10} + \frac{118833}{2209943} a^{9} - \frac{788282}{2209943} a^{8} - \frac{403544}{2209943} a^{7} + \frac{1036290}{2209943} a^{6} + \frac{262288}{2209943} a^{5} - \frac{564947}{2209943} a^{4} - \frac{1079307}{2209943} a^{3} - \frac{444679}{2209943} a^{2} - \frac{452947}{2209943} a - \frac{745664}{2209943}$, $\frac{1}{2209943} a^{31} - \frac{1017266}{2209943} a^{21} + \frac{145529}{2209943} a^{20} + \frac{23074}{2209943} a^{19} - \frac{728282}{2209943} a^{18} - \frac{85634}{2209943} a^{17} - \frac{571634}{2209943} a^{16} + \frac{172326}{2209943} a^{15} + \frac{105380}{2209943} a^{14} - \frac{647993}{2209943} a^{13} + \frac{550179}{2209943} a^{12} + \frac{819898}{2209943} a^{11} + \frac{364855}{2209943} a^{10} + \frac{870146}{2209943} a^{9} - \frac{825296}{2209943} a^{8} - \frac{588768}{2209943} a^{7} + \frac{684040}{2209943} a^{6} + \frac{237613}{2209943} a^{5} - \frac{445612}{2209943} a^{4} - \frac{250825}{2209943} a^{3} - \frac{579686}{2209943} a^{2} + \frac{760058}{2209943} a - \frac{616786}{2209943}$, $\frac{1}{2209943} a^{32} + \frac{489397}{2209943} a^{21} - \frac{537770}{2209943} a^{20} + \frac{298863}{2209943} a^{19} - \frac{479313}{2209943} a^{18} + \frac{829243}{2209943} a^{17} + \frac{677186}{2209943} a^{16} - \frac{10853}{2209943} a^{15} - \frac{330488}{2209943} a^{14} - \frac{681861}{2209943} a^{13} + \frac{760196}{2209943} a^{12} + \frac{749925}{2209943} a^{11} - \frac{70645}{2209943} a^{10} + \frac{688914}{2209943} a^{9} - \frac{238808}{2209943} a^{8} + \frac{804324}{2209943} a^{7} - \frac{112347}{2209943} a^{6} + \frac{137218}{2209943} a^{5} - \frac{808006}{2209943} a^{4} - \frac{468150}{2209943} a^{3} - \frac{12483}{2209943} a^{2} - \frac{548592}{2209943} a - \frac{978981}{2209943}$, $\frac{1}{2209943} a^{33} - \frac{166176}{2209943} a^{21} - \frac{267099}{2209943} a^{20} - \frac{616660}{2209943} a^{19} + \frac{532806}{2209943} a^{18} + \frac{939489}{2209943} a^{17} - \frac{550065}{2209943} a^{16} + \frac{546591}{2209943} a^{15} - \frac{33705}{2209943} a^{14} - \frac{107174}{2209943} a^{13} - \frac{387750}{2209943} a^{12} + \frac{1102155}{2209943} a^{11} - \frac{49880}{2209943} a^{10} - \frac{404611}{2209943} a^{9} - \frac{139115}{2209943} a^{8} - \frac{274626}{2209943} a^{7} + \frac{1080657}{2209943} a^{6} + \frac{428037}{2209943} a^{5} - \frac{158652}{2209943} a^{4} - \frac{61028}{2209943} a^{3} - \frac{168503}{2209943} a^{2} + \frac{834450}{2209943} a + \frac{151811}{2209943}$, $\frac{1}{2209943} a^{34} + \frac{696382}{2209943} a^{21} + \frac{911368}{2209943} a^{20} - \frac{154422}{2209943} a^{19} + \frac{636216}{2209943} a^{18} + \frac{770324}{2209943} a^{17} + \frac{662340}{2209943} a^{16} - \frac{894241}{2209943} a^{15} - \frac{878834}{2209943} a^{14} + \frac{247446}{2209943} a^{13} + \frac{896766}{2209943} a^{12} - \frac{759190}{2209943} a^{11} - \frac{392277}{2209943} a^{10} + \frac{612557}{2209943} a^{9} - \frac{295770}{2209943} a^{8} + \frac{111462}{2209943} a^{7} + \frac{449181}{2209943} a^{6} - \frac{990799}{2209943} a^{5} + \frac{778107}{2209943} a^{4} + \frac{958197}{2209943} a^{3} + \frac{666623}{2209943} a^{2} + \frac{631610}{2209943} a + \frac{5548}{2209943}$, $\frac{1}{2209943} a^{35} - \frac{1070733}{2209943} a^{21} - \frac{452816}{2209943} a^{20} - \frac{93376}{2209943} a^{19} - \frac{190019}{2209943} a^{18} + \frac{535338}{2209943} a^{17} + \frac{13}{257} a^{16} + \frac{891692}{2209943} a^{15} - \frac{566226}{2209943} a^{14} + \frac{360896}{2209943} a^{13} + \frac{619295}{2209943} a^{12} - \frac{512327}{2209943} a^{11} - \frac{415848}{2209943} a^{10} + \frac{488784}{2209943} a^{9} - \frac{335501}{2209943} a^{8} + \frac{17981}{2209943} a^{7} - \frac{543836}{2209943} a^{6} - \frac{233581}{2209943} a^{5} + \frac{672320}{2209943} a^{4} - \frac{1086771}{2209943} a^{3} - \frac{1079169}{2209943} a^{2} + \frac{76285}{2209943} a - \frac{655741}{2209943}$, $\frac{1}{2209943} a^{36} - \frac{118825}{2209943} a^{21} - \frac{796825}{2209943} a^{20} - \frac{957983}{2209943} a^{19} + \frac{641211}{2209943} a^{18} + \frac{675481}{2209943} a^{17} + \frac{1034391}{2209943} a^{16} - \frac{597289}{2209943} a^{15} + \frac{146217}{2209943} a^{14} + \frac{1071148}{2209943} a^{13} - \frac{164981}{2209943} a^{12} - \frac{189126}{2209943} a^{11} + \frac{638554}{2209943} a^{10} + \frac{583585}{2209943} a^{9} + \frac{1024057}{2209943} a^{8} - \frac{315799}{2209943} a^{7} + \frac{970286}{2209943} a^{6} - \frac{276844}{2209943} a^{5} - \frac{684165}{2209943} a^{4} - \frac{973176}{2209943} a^{3} + \frac{879741}{2209943} a^{2} - \frac{828027}{2209943} a + \frac{1014487}{2209943}$, $\frac{1}{2209943} a^{37} - \frac{788024}{2209943} a^{21} + \frac{443546}{2209943} a^{20} + \frac{271569}{2209943} a^{19} + \frac{565746}{2209943} a^{18} + \frac{1058118}{2209943} a^{17} + \frac{231980}{2209943} a^{16} - \frac{583909}{2209943} a^{15} - \frac{37426}{2209943} a^{14} + \frac{574718}{2209943} a^{13} + \frac{996271}{2209943} a^{12} - \frac{699312}{2209943} a^{11} + \frac{86172}{2209943} a^{10} + \frac{856707}{2209943} a^{9} + \frac{536909}{2209943} a^{8} + \frac{494574}{2209943} a^{7} + \frac{1080391}{2209943} a^{6} + \frac{550042}{2209943} a^{5} - \frac{25903}{2209943} a^{4} + \frac{831876}{2209943} a^{3} - \frac{575493}{2209943} a^{2} + \frac{216093}{2209943} a - \frac{519740}{2209943}$, $\frac{1}{2209943} a^{38} + \frac{1001872}{2209943} a^{21} + \frac{957601}{2209943} a^{20} - \frac{784516}{2209943} a^{19} - \frac{926775}{2209943} a^{18} + \frac{227323}{2209943} a^{17} + \frac{140210}{2209943} a^{16} - \frac{317910}{2209943} a^{15} + \frac{167144}{2209943} a^{14} - \frac{521244}{2209943} a^{13} - \frac{825888}{2209943} a^{12} + \frac{479623}{2209943} a^{11} - \frac{1005152}{2209943} a^{10} + \frac{802886}{2209943} a^{9} + \frac{844977}{2209943} a^{8} + \frac{193458}{2209943} a^{7} + \frac{199639}{2209943} a^{6} - \frac{1094488}{2209943} a^{5} + \frac{501583}{2209943} a^{4} - \frac{425270}{2209943} a^{3} - \frac{159837}{2209943} a^{2} - \frac{897755}{2209943} a + \frac{980419}{2209943}$, $\frac{1}{2209943} a^{39} + \frac{1023887}{2209943} a^{21} - \frac{901008}{2209943} a^{20} + \frac{709099}{2209943} a^{19} + \frac{230728}{2209943} a^{18} - \frac{576263}{2209943} a^{17} + \frac{227095}{2209943} a^{16} - \frac{800028}{2209943} a^{15} + \frac{1001923}{2209943} a^{14} - \frac{962021}{2209943} a^{13} + \frac{482716}{2209943} a^{12} - \frac{253239}{2209943} a^{11} - \frac{863077}{2209943} a^{10} + \frac{725812}{2209943} a^{9} - \frac{423373}{2209943} a^{8} + \frac{961050}{2209943} a^{7} - \frac{477657}{2209943} a^{6} - \frac{28405}{2209943} a^{5} - \frac{854924}{2209943} a^{4} + \frac{936802}{2209943} a^{3} + \frac{72153}{2209943} a^{2} + \frac{981328}{2209943} a - \frac{196439}{2209943}$, $\frac{1}{2209943} a^{40} + \frac{415406}{2209943} a^{21} - \frac{648676}{2209943} a^{20} + \frac{189915}{2209943} a^{19} + \frac{912089}{2209943} a^{18} + \frac{961729}{2209943} a^{17} + \frac{666690}{2209943} a^{16} - \frac{505653}{2209943} a^{15} + \frac{309574}{2209943} a^{14} - \frac{215527}{2209943} a^{13} - \frac{758785}{2209943} a^{12} - \frac{128878}{2209943} a^{11} + \frac{1011092}{2209943} a^{10} + \frac{728509}{2209943} a^{9} - \frac{790826}{2209943} a^{8} + \frac{1031944}{2209943} a^{7} - \frac{486472}{2209943} a^{6} - \frac{799250}{2209943} a^{5} - \frac{528248}{2209943} a^{4} + \frac{51533}{2209943} a^{3} - \frac{935605}{2209943} a^{2} + \frac{810330}{2209943} a - \frac{630662}{2209943}$, $\frac{1}{2209943} a^{41} - \frac{73753}{2209943} a^{21} + \frac{788235}{2209943} a^{20} + \frac{1074696}{2209943} a^{19} + \frac{643364}{2209943} a^{18} - \frac{50293}{2209943} a^{17} + \frac{4762}{2209943} a^{16} - \frac{406911}{2209943} a^{15} - \frac{671703}{2209943} a^{14} + \frac{546644}{2209943} a^{13} - \frac{593309}{2209943} a^{12} - \frac{323178}{2209943} a^{11} + \frac{348237}{2209943} a^{10} - \frac{721417}{2209943} a^{9} - \frac{210398}{2209943} a^{8} - \frac{859066}{2209943} a^{7} + \frac{443092}{2209943} a^{6} + \frac{792667}{2209943} a^{5} + \frac{94417}{2209943} a^{4} + \frac{539561}{2209943} a^{3} - \frac{82224}{2209943} a^{2} + \frac{92388}{2209943} a - \frac{153134}{2209943}$

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: $41$ 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);

## Galois group

$C_{42}$ (as 42T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 42 The 42 conjugacy class representatives for $C_{42}$ Character table for $C_{42}$ is not computed

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.14.0.1}{14} }^{3}$ $21^{2}$ $21^{2}$ R ${\href{/LocalNumberField/11.7.0.1}{7} }^{6}$ $42$ $42$ $21^{2}$ $21^{2}$ $42$ $42$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }^{3}$ R ${\href{/LocalNumberField/47.14.0.1}{14} }^{3}$ $21^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }^{3}$

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
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 7.6.3.1x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 7.6.3.1x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3} 7.6.3.1x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
43Data not computed