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Properties

Label 42.0.896...984.1
Degree $42$
Signature $[0, 21]$
Discriminant $-8.961\times 10^{88}$
Root discriminant \(131.19\)
Ramified primes $2,7,29$
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 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649)
 
gp: K = bnfinit(y^42 + 108*y^40 + 5156*y^38 + 144488*y^36 + 2662832*y^34 + 34265834*y^32 + 318981566*y^30 + 2195598458*y^28 + 11324272470*y^26 + 44091529459*y^24 + 129986543401*y^22 + 290044496138*y^20 + 488271862609*y^18 + 616608189604*y^16 + 579268843957*y^14 + 400104193619*y^12 + 199814257891*y^10 + 70396440790*y^8 + 16842157767*y^6 + 2568252188*y^4 + 221604496*y^2 + 8082649, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649)
 

\( x^{42} + 108 x^{40} + 5156 x^{38} + 144488 x^{36} + 2662832 x^{34} + 34265834 x^{32} + 318981566 x^{30} + \cdots + 8082649 \) 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:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-896\!\cdots\!984\) \(\medspace = -\,2^{42}\cdot 7^{28}\cdot 29^{36}\) 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:  \(131.19\)
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:  $2\cdot 7^{2/3}29^{6/7}\approx 131.19361141265577$
Ramified primes:   \(2\), \(7\), \(29\) 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:  \(\Q(\sqrt{-1}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(812=2^{2}\cdot 7\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{812}(1,·)$, $\chi_{812}(645,·)$, $\chi_{812}(513,·)$, $\chi_{812}(393,·)$, $\chi_{812}(779,·)$, $\chi_{812}(141,·)$, $\chi_{812}(401,·)$, $\chi_{812}(529,·)$, $\chi_{812}(277,·)$, $\chi_{812}(23,·)$, $\chi_{812}(25,·)$, $\chi_{812}(687,·)$, $\chi_{812}(799,·)$, $\chi_{812}(291,·)$, $\chi_{812}(165,·)$, $\chi_{812}(807,·)$, $\chi_{812}(169,·)$, $\chi_{812}(683,·)$, $\chi_{812}(429,·)$, $\chi_{812}(431,·)$, $\chi_{812}(53,·)$, $\chi_{812}(697,·)$, $\chi_{812}(571,·)$, $\chi_{812}(575,·)$, $\chi_{812}(65,·)$, $\chi_{812}(603,·)$, $\chi_{812}(407,·)$, $\chi_{812}(197,·)$, $\chi_{812}(459,·)$, $\chi_{812}(81,·)$, $\chi_{812}(547,·)$, $\chi_{812}(471,·)$, $\chi_{812}(219,·)$, $\chi_{812}(487,·)$, $\chi_{812}(233,·)$, $\chi_{812}(107,·)$, $\chi_{812}(239,·)$, $\chi_{812}(625,·)$, $\chi_{812}(373,·)$, $\chi_{812}(123,·)$, $\chi_{812}(281,·)$, $\chi_{812}(639,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{1048576}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{41}a^{28}-\frac{2}{41}a^{26}+\frac{12}{41}a^{24}-\frac{12}{41}a^{22}+\frac{16}{41}a^{20}+\frac{11}{41}a^{18}+\frac{6}{41}a^{16}+\frac{2}{41}a^{14}-\frac{10}{41}a^{12}-\frac{16}{41}a^{10}-\frac{10}{41}a^{8}-\frac{2}{41}a^{6}-\frac{12}{41}a^{4}-\frac{17}{41}a^{2}+\frac{2}{41}$, $\frac{1}{41}a^{29}-\frac{2}{41}a^{27}+\frac{12}{41}a^{25}-\frac{12}{41}a^{23}+\frac{16}{41}a^{21}+\frac{11}{41}a^{19}+\frac{6}{41}a^{17}+\frac{2}{41}a^{15}-\frac{10}{41}a^{13}-\frac{16}{41}a^{11}-\frac{10}{41}a^{9}-\frac{2}{41}a^{7}-\frac{12}{41}a^{5}-\frac{17}{41}a^{3}+\frac{2}{41}a$, $\frac{1}{41}a^{30}+\frac{8}{41}a^{26}+\frac{12}{41}a^{24}-\frac{8}{41}a^{22}+\frac{2}{41}a^{20}-\frac{13}{41}a^{18}+\frac{14}{41}a^{16}-\frac{6}{41}a^{14}+\frac{5}{41}a^{12}-\frac{1}{41}a^{10}+\frac{19}{41}a^{8}-\frac{16}{41}a^{6}+\frac{9}{41}a^{2}+\frac{4}{41}$, $\frac{1}{41}a^{31}+\frac{8}{41}a^{27}+\frac{12}{41}a^{25}-\frac{8}{41}a^{23}+\frac{2}{41}a^{21}-\frac{13}{41}a^{19}+\frac{14}{41}a^{17}-\frac{6}{41}a^{15}+\frac{5}{41}a^{13}-\frac{1}{41}a^{11}+\frac{19}{41}a^{9}-\frac{16}{41}a^{7}+\frac{9}{41}a^{3}+\frac{4}{41}a$, $\frac{1}{41}a^{32}-\frac{13}{41}a^{26}+\frac{19}{41}a^{24}+\frac{16}{41}a^{22}-\frac{18}{41}a^{20}+\frac{8}{41}a^{18}-\frac{13}{41}a^{16}-\frac{11}{41}a^{14}-\frac{3}{41}a^{12}-\frac{17}{41}a^{10}-\frac{18}{41}a^{8}+\frac{16}{41}a^{6}-\frac{18}{41}a^{4}+\frac{17}{41}a^{2}-\frac{16}{41}$, $\frac{1}{41}a^{33}-\frac{13}{41}a^{27}+\frac{19}{41}a^{25}+\frac{16}{41}a^{23}-\frac{18}{41}a^{21}+\frac{8}{41}a^{19}-\frac{13}{41}a^{17}-\frac{11}{41}a^{15}-\frac{3}{41}a^{13}-\frac{17}{41}a^{11}-\frac{18}{41}a^{9}+\frac{16}{41}a^{7}-\frac{18}{41}a^{5}+\frac{17}{41}a^{3}-\frac{16}{41}a$, $\frac{1}{41}a^{34}-\frac{7}{41}a^{26}+\frac{8}{41}a^{24}-\frac{10}{41}a^{22}+\frac{11}{41}a^{20}+\frac{7}{41}a^{18}-\frac{15}{41}a^{16}-\frac{18}{41}a^{14}+\frac{17}{41}a^{12}+\frac{20}{41}a^{10}+\frac{9}{41}a^{8}-\frac{3}{41}a^{6}-\frac{16}{41}a^{4}+\frac{9}{41}a^{2}-\frac{15}{41}$, $\frac{1}{41}a^{35}-\frac{7}{41}a^{27}+\frac{8}{41}a^{25}-\frac{10}{41}a^{23}+\frac{11}{41}a^{21}+\frac{7}{41}a^{19}-\frac{15}{41}a^{17}-\frac{18}{41}a^{15}+\frac{17}{41}a^{13}+\frac{20}{41}a^{11}+\frac{9}{41}a^{9}-\frac{3}{41}a^{7}-\frac{16}{41}a^{5}+\frac{9}{41}a^{3}-\frac{15}{41}a$, $\frac{1}{697}a^{36}-\frac{3}{697}a^{34}-\frac{2}{697}a^{32}+\frac{7}{697}a^{30}+\frac{5}{697}a^{28}+\frac{128}{697}a^{26}+\frac{156}{697}a^{24}-\frac{150}{697}a^{22}+\frac{11}{697}a^{20}-\frac{216}{697}a^{18}-\frac{146}{697}a^{16}+\frac{157}{697}a^{14}-\frac{233}{697}a^{12}-\frac{298}{697}a^{10}-\frac{309}{697}a^{8}+\frac{9}{41}a^{6}-\frac{10}{697}a^{4}-\frac{299}{697}a^{2}-\frac{322}{697}$, $\frac{1}{697}a^{37}-\frac{3}{697}a^{35}-\frac{2}{697}a^{33}+\frac{7}{697}a^{31}+\frac{5}{697}a^{29}+\frac{128}{697}a^{27}+\frac{156}{697}a^{25}-\frac{150}{697}a^{23}+\frac{11}{697}a^{21}-\frac{216}{697}a^{19}-\frac{146}{697}a^{17}+\frac{157}{697}a^{15}-\frac{233}{697}a^{13}-\frac{298}{697}a^{11}-\frac{309}{697}a^{9}+\frac{9}{41}a^{7}-\frac{10}{697}a^{5}-\frac{299}{697}a^{3}-\frac{322}{697}a$, $\frac{1}{187826832067837}a^{38}-\frac{50723618660}{187826832067837}a^{36}+\frac{1221832958179}{187826832067837}a^{34}+\frac{510713146435}{187826832067837}a^{32}+\frac{1608511785163}{187826832067837}a^{30}+\frac{774026028124}{187826832067837}a^{28}-\frac{44462739652157}{187826832067837}a^{26}-\frac{9249828091152}{187826832067837}a^{24}+\frac{84424648732937}{187826832067837}a^{22}+\frac{32995320682301}{187826832067837}a^{20}-\frac{10403622782925}{187826832067837}a^{18}+\frac{92644711871534}{187826832067837}a^{16}-\frac{17620678824946}{187826832067837}a^{14}-\frac{59400134441736}{187826832067837}a^{12}-\frac{86652089205468}{187826832067837}a^{10}-\frac{61428513121647}{187826832067837}a^{8}-\frac{23970126732222}{187826832067837}a^{6}+\frac{72070737010553}{187826832067837}a^{4}-\frac{77245390807947}{187826832067837}a^{2}+\frac{26125422954385}{187826832067837}$, $\frac{1}{187826832067837}a^{39}-\frac{50723618660}{187826832067837}a^{37}+\frac{1221832958179}{187826832067837}a^{35}+\frac{510713146435}{187826832067837}a^{33}+\frac{1608511785163}{187826832067837}a^{31}+\frac{774026028124}{187826832067837}a^{29}-\frac{44462739652157}{187826832067837}a^{27}-\frac{9249828091152}{187826832067837}a^{25}+\frac{84424648732937}{187826832067837}a^{23}+\frac{32995320682301}{187826832067837}a^{21}-\frac{10403622782925}{187826832067837}a^{19}+\frac{92644711871534}{187826832067837}a^{17}-\frac{17620678824946}{187826832067837}a^{15}-\frac{59400134441736}{187826832067837}a^{13}-\frac{86652089205468}{187826832067837}a^{11}-\frac{61428513121647}{187826832067837}a^{9}-\frac{23970126732222}{187826832067837}a^{7}+\frac{72070737010553}{187826832067837}a^{5}-\frac{77245390807947}{187826832067837}a^{3}+\frac{26125422954385}{187826832067837}a$, $\frac{1}{50\!\cdots\!63}a^{40}-\frac{12\!\cdots\!74}{50\!\cdots\!63}a^{38}+\frac{18\!\cdots\!42}{50\!\cdots\!63}a^{36}+\frac{72\!\cdots\!86}{29\!\cdots\!39}a^{34}+\frac{10\!\cdots\!20}{50\!\cdots\!63}a^{32}+\frac{55\!\cdots\!52}{50\!\cdots\!63}a^{30}-\frac{52\!\cdots\!48}{50\!\cdots\!63}a^{28}-\frac{21\!\cdots\!84}{50\!\cdots\!63}a^{26}+\frac{93\!\cdots\!41}{50\!\cdots\!63}a^{24}-\frac{14\!\cdots\!13}{50\!\cdots\!63}a^{22}-\frac{10\!\cdots\!22}{50\!\cdots\!63}a^{20}-\frac{10\!\cdots\!58}{50\!\cdots\!63}a^{18}-\frac{10\!\cdots\!69}{50\!\cdots\!63}a^{16}-\frac{10\!\cdots\!07}{50\!\cdots\!63}a^{14}+\frac{10\!\cdots\!36}{50\!\cdots\!63}a^{12}+\frac{69\!\cdots\!92}{50\!\cdots\!63}a^{10}-\frac{11\!\cdots\!06}{50\!\cdots\!63}a^{8}-\frac{82\!\cdots\!67}{50\!\cdots\!63}a^{6}+\frac{24\!\cdots\!07}{50\!\cdots\!63}a^{4}+\frac{17\!\cdots\!94}{50\!\cdots\!63}a^{2}-\frac{66\!\cdots\!25}{50\!\cdots\!63}$, $\frac{1}{14\!\cdots\!09}a^{41}-\frac{37\!\cdots\!71}{14\!\cdots\!09}a^{39}-\frac{18\!\cdots\!71}{14\!\cdots\!09}a^{37}+\frac{15\!\cdots\!80}{14\!\cdots\!09}a^{35}-\frac{22\!\cdots\!79}{14\!\cdots\!09}a^{33}-\frac{10\!\cdots\!13}{14\!\cdots\!09}a^{31}+\frac{51\!\cdots\!18}{14\!\cdots\!09}a^{29}+\frac{51\!\cdots\!11}{14\!\cdots\!09}a^{27}+\frac{29\!\cdots\!76}{14\!\cdots\!09}a^{25}-\frac{45\!\cdots\!33}{14\!\cdots\!09}a^{23}+\frac{54\!\cdots\!29}{14\!\cdots\!09}a^{21}+\frac{54\!\cdots\!69}{14\!\cdots\!09}a^{19}-\frac{60\!\cdots\!20}{14\!\cdots\!09}a^{17}-\frac{15\!\cdots\!76}{83\!\cdots\!77}a^{15}+\frac{46\!\cdots\!11}{14\!\cdots\!09}a^{13}-\frac{45\!\cdots\!94}{14\!\cdots\!09}a^{11}-\frac{52\!\cdots\!44}{14\!\cdots\!09}a^{9}-\frac{50\!\cdots\!48}{14\!\cdots\!09}a^{7}-\frac{13\!\cdots\!93}{14\!\cdots\!09}a^{5}-\frac{70\!\cdots\!25}{14\!\cdots\!09}a^{3}+\frac{11\!\cdots\!94}{14\!\cdots\!09}a$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $41$

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:  $20$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{2079519440197074364894543310912036}{617606824923260580261289005004530509} a^{41} - \frac{223792148505866147537082121599344301}{617606824923260580261289005004530509} a^{39} - \frac{10636344041198208571666816465416386860}{617606824923260580261289005004530509} a^{37} - \frac{296394453047644195643303005771952963837}{617606824923260580261289005004530509} a^{35} - \frac{5423963113863009604822895672333752239734}{617606824923260580261289005004530509} a^{33} - \frac{69180388157686556054042142415302571287068}{617606824923260580261289005004530509} a^{31} - \frac{636848736631251001186852339750665602939765}{617606824923260580261289005004530509} a^{29} - \frac{4322026997456446865556448691305118563200849}{617606824923260580261289005004530509} a^{27} - \frac{21894719750636794813227130896379825276415012}{617606824923260580261289005004530509} a^{25} - \frac{83308580934558081364287326503066953732978688}{617606824923260580261289005004530509} a^{23} - \frac{238421367470295726634472432867311376715621166}{617606824923260580261289005004530509} a^{21} - \frac{511891268852295488217807387920234964099740529}{617606824923260580261289005004530509} a^{19} - \frac{819428936512667177149992781397757308749218786}{617606824923260580261289005004530509} a^{17} - \frac{968582989840749820109151945190578012552926028}{617606824923260580261289005004530509} a^{15} - \frac{833834967171972015630785246764628319299537276}{617606824923260580261289005004530509} a^{13} - \frac{512833263667997644307112189948970515856585038}{617606824923260580261289005004530509} a^{11} - \frac{219201824971949301496786614413528476188460919}{617606824923260580261289005004530509} a^{9} - \frac{62477354549333550624539346301378908190900309}{617606824923260580261289005004530509} a^{7} - \frac{11106023942762739321171474383039323264278006}{617606824923260580261289005004530509} a^{5} - \frac{1089072856117519150253481080449568849408350}{617606824923260580261289005004530509} a^{3} - \frac{43906924214563157399060034454102838747056}{617606824923260580261289005004530509} a \)  (order $4$) 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 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649)
 
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 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649, 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 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649);
 
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 + 108*x^40 + 5156*x^38 + 144488*x^36 + 2662832*x^34 + 34265834*x^32 + 318981566*x^30 + 2195598458*x^28 + 11324272470*x^26 + 44091529459*x^24 + 129986543401*x^22 + 290044496138*x^20 + 488271862609*x^18 + 616608189604*x^16 + 579268843957*x^14 + 400104193619*x^12 + 199814257891*x^10 + 70396440790*x^8 + 16842157767*x^6 + 2568252188*x^4 + 221604496*x^2 + 8082649);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\zeta_{7})^+\), 6.0.153664.1, 7.7.594823321.1, 14.0.5796901408038404767744.1, 21.21.142736986105602839685204351151303673689.1

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 R $42$ $21^{2}$ R $42$ ${\href{/padicField/13.7.0.1}{7} }^{6}$ ${\href{/padicField/17.3.0.1}{3} }^{14}$ $42$ $42$ R $42$ $21^{2}$ ${\href{/padicField/41.1.0.1}{1} }^{42}$ ${\href{/padicField/43.14.0.1}{14} }^{3}$ $42$ $21^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{7}$

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
\(2\) Copy content Toggle raw display Deg $42$$2$$21$$42$
\(7\) Copy content Toggle raw display Deg $42$$3$$14$$28$
\(29\) Copy content Toggle raw display 29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.2$x^{7} + 29$$7$$1$$6$$C_7$$[\ ]_{7}$