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Properties

Label 42.42.523...301.1
Degree $42$
Signature $[42, 0]$
Discriminant $5.239\times 10^{84}$
Root discriminant \(104.02\)
Ramified primes $7,43$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

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

\( x^{42} - x^{41} - 85 x^{40} + 85 x^{39} + 3355 x^{38} - 3355 x^{37} - 81613 x^{36} + 81613 x^{35} + \cdots - 998717 \) Copy content Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(523\!\cdots\!301\) \(\medspace = 7^{21}\cdot 43^{41}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(104.02\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $7^{1/2}43^{41/42}\approx 104.02204070198826$
Ramified primes:   \(7\), \(43\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{301}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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 \(a\))

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $41$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{301}) \), 3.3.1849.1, 6.6.50423895949.1, 7.7.6321363049.1, 14.14.1415064391703810600151375949.1, \(\Q(\zeta_{43})^+\)

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Frobenius cycle types

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

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display 7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
\(43\) Copy content Toggle raw display Deg $42$$42$$1$$41$