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Properties

Label 42.0.233...507.1
Degree $42$
Signature $[0, 21]$
Discriminant $-2.334\times 10^{87}$
Root discriminant \(120.28\)
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 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947)
 
gp: K = bnfinit(y^42 - 11*y^41 + 31*y^40 + 15*y^39 + 149*y^38 - 2171*y^37 + 5622*y^36 - 5813*y^35 + 17703*y^34 - 77949*y^33 + 215171*y^32 - 500434*y^31 + 1078714*y^30 - 3010069*y^29 + 7913255*y^28 - 11776682*y^27 + 16679313*y^26 - 87643658*y^25 + 355007978*y^24 - 761687128*y^23 + 908129906*y^22 - 907038305*y^21 + 2012009714*y^20 - 6104907919*y^19 + 18723809190*y^18 - 48743446573*y^17 + 87856746678*y^16 - 105007899193*y^15 + 97597592982*y^14 - 99989052476*y^13 + 101465465536*y^12 - 67260548547*y^11 + 47339909902*y^10 - 82545626769*y^9 + 101683114718*y^8 - 83757841269*y^7 + 103258810875*y^6 - 130872100909*y^5 + 93586325459*y^4 - 47886183228*y^3 + 41077962943*y^2 - 30874848118*y + 9042834947, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947)
 

\( x^{42} - 11 x^{41} + 31 x^{40} + 15 x^{39} + 149 x^{38} - 2171 x^{37} + 5622 x^{36} - 5813 x^{35} + \cdots + 9042834947 \) 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:   \(-233\!\cdots\!507\) \(\medspace = -\,7^{28}\cdot 43^{39}\) 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:  \(120.28\)
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:  $7^{2/3}43^{13/14}\approx 120.2792830317974$
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:  \(\Q(\sqrt{-43}) \)
$\card{ \Gal(K/\Q) }$:  $42$
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$$\chi_{301}(128,·)$, $\chi_{301}(1,·)$, $\chi_{301}(2,·)$, $\chi_{301}(4,·)$, $\chi_{301}(65,·)$, $\chi_{301}(8,·)$, $\chi_{301}(137,·)$, $\chi_{301}(11,·)$, $\chi_{301}(130,·)$, $\chi_{301}(256,·)$, $\chi_{301}(16,·)$, $\chi_{301}(274,·)$, $\chi_{301}(22,·)$, $\chi_{301}(151,·)$, $\chi_{301}(260,·)$, $\chi_{301}(156,·)$, $\chi_{301}(32,·)$, $\chi_{301}(39,·)$, $\chi_{301}(170,·)$, $\chi_{301}(44,·)$, $\chi_{301}(176,·)$, $\chi_{301}(51,·)$, $\chi_{301}(183,·)$, $\chi_{301}(64,·)$, $\chi_{301}(193,·)$, $\chi_{301}(204,·)$, $\chi_{301}(78,·)$, $\chi_{301}(207,·)$, $\chi_{301}(211,·)$, $\chi_{301}(85,·)$, $\chi_{301}(214,·)$, $\chi_{301}(88,·)$, $\chi_{301}(219,·)$, $\chi_{301}(226,·)$, $\chi_{301}(102,·)$, $\chi_{301}(107,·)$, $\chi_{301}(113,·)$, $\chi_{301}(242,·)$, $\chi_{301}(247,·)$, $\chi_{301}(121,·)$, $\chi_{301}(254,·)$, $\chi_{301}(127,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{1048576}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $\frac{1}{337}a^{37}+\frac{83}{337}a^{36}+\frac{119}{337}a^{35}-\frac{44}{337}a^{34}-\frac{149}{337}a^{33}+\frac{41}{337}a^{32}-\frac{157}{337}a^{31}+\frac{73}{337}a^{30}+\frac{167}{337}a^{29}+\frac{106}{337}a^{28}-\frac{140}{337}a^{27}-\frac{167}{337}a^{26}+\frac{150}{337}a^{25}+\frac{137}{337}a^{24}+\frac{83}{337}a^{23}-\frac{35}{337}a^{22}+\frac{141}{337}a^{21}+\frac{47}{337}a^{20}+\frac{114}{337}a^{19}+\frac{113}{337}a^{18}+\frac{75}{337}a^{17}-\frac{91}{337}a^{16}+\frac{166}{337}a^{15}+\frac{41}{337}a^{14}-\frac{152}{337}a^{13}+\frac{117}{337}a^{12}-\frac{23}{337}a^{11}-\frac{46}{337}a^{10}-\frac{123}{337}a^{9}+\frac{96}{337}a^{8}+\frac{94}{337}a^{7}+\frac{117}{337}a^{6}-\frac{47}{337}a^{5}-\frac{145}{337}a^{4}-\frac{35}{337}a^{3}+\frac{110}{337}a^{2}-\frac{129}{337}a+\frac{71}{337}$, $\frac{1}{84587}a^{38}-\frac{22}{84587}a^{37}-\frac{10281}{84587}a^{36}+\frac{4985}{84587}a^{35}+\frac{32442}{84587}a^{34}+\frac{7598}{84587}a^{33}-\frac{22323}{84587}a^{32}-\frac{14446}{84587}a^{31}+\frac{24854}{84587}a^{30}-\frac{17092}{84587}a^{29}-\frac{42274}{84587}a^{28}+\frac{8467}{84587}a^{27}+\frac{5216}{84587}a^{26}+\frac{9325}{84587}a^{25}+\frac{5244}{84587}a^{24}-\frac{36721}{84587}a^{23}-\frac{22470}{84587}a^{22}+\frac{17257}{84587}a^{21}+\frac{34608}{84587}a^{20}-\frac{17923}{84587}a^{19}+\frac{20899}{84587}a^{18}+\frac{39214}{84587}a^{17}+\frac{14439}{84587}a^{16}+\frac{38890}{84587}a^{15}+\frac{28906}{84587}a^{14}+\frac{20121}{84587}a^{13}-\frac{5568}{84587}a^{12}+\frac{15512}{84587}a^{11}-\frac{31015}{84587}a^{10}+\frac{7956}{84587}a^{9}+\frac{13604}{84587}a^{8}+\frac{28328}{84587}a^{7}-\frac{22779}{84587}a^{6}+\frac{11193}{84587}a^{5}-\frac{33675}{84587}a^{4}-\frac{933}{84587}a^{3}-\frac{20778}{84587}a^{2}+\frac{22715}{84587}a+\frac{14787}{84587}$, $\frac{1}{17\!\cdots\!49}a^{39}+\frac{6002678012}{17\!\cdots\!49}a^{38}+\frac{1261619063676}{17\!\cdots\!49}a^{37}+\frac{407043669912976}{17\!\cdots\!49}a^{36}+\frac{2819756335677}{17\!\cdots\!49}a^{35}-\frac{355293810027673}{17\!\cdots\!49}a^{34}-\frac{87455754300957}{17\!\cdots\!49}a^{33}+\frac{808897253512992}{17\!\cdots\!49}a^{32}-\frac{258991936282850}{17\!\cdots\!49}a^{31}+\frac{426966805727161}{17\!\cdots\!49}a^{30}-\frac{237512242123769}{17\!\cdots\!49}a^{29}+\frac{557763975630923}{17\!\cdots\!49}a^{28}+\frac{320451456512828}{17\!\cdots\!49}a^{27}-\frac{618538654617230}{17\!\cdots\!49}a^{26}+\frac{5066411768737}{17\!\cdots\!49}a^{25}+\frac{366576915917628}{17\!\cdots\!49}a^{24}-\frac{187488780099930}{17\!\cdots\!49}a^{23}+\frac{804708238194396}{17\!\cdots\!49}a^{22}+\frac{354795532741723}{17\!\cdots\!49}a^{21}-\frac{869287891762369}{17\!\cdots\!49}a^{20}-\frac{225587811249473}{17\!\cdots\!49}a^{19}+\frac{128430945426788}{17\!\cdots\!49}a^{18}+\frac{395079880039544}{17\!\cdots\!49}a^{17}+\frac{504514888289626}{17\!\cdots\!49}a^{16}-\frac{821956349092599}{17\!\cdots\!49}a^{15}-\frac{738953494198687}{17\!\cdots\!49}a^{14}-\frac{755660907837977}{17\!\cdots\!49}a^{13}-\frac{854105875346853}{17\!\cdots\!49}a^{12}+\frac{555191232186713}{17\!\cdots\!49}a^{11}+\frac{675984304095620}{17\!\cdots\!49}a^{10}+\frac{252622714445955}{17\!\cdots\!49}a^{9}-\frac{854986080701802}{17\!\cdots\!49}a^{8}-\frac{558655499254597}{17\!\cdots\!49}a^{7}-\frac{681540335262451}{17\!\cdots\!49}a^{6}+\frac{335373309336563}{17\!\cdots\!49}a^{5}-\frac{692466245559277}{17\!\cdots\!49}a^{4}+\frac{14901982856660}{17\!\cdots\!49}a^{3}+\frac{138969006411012}{17\!\cdots\!49}a^{2}-\frac{831437809048532}{17\!\cdots\!49}a+\frac{161158348300817}{17\!\cdots\!49}$, $\frac{1}{94\!\cdots\!31}a^{40}+\frac{384}{94\!\cdots\!31}a^{39}+\frac{8315592999579}{94\!\cdots\!31}a^{38}+\frac{13\!\cdots\!58}{94\!\cdots\!31}a^{37}-\frac{32\!\cdots\!88}{94\!\cdots\!31}a^{36}+\frac{12\!\cdots\!84}{94\!\cdots\!31}a^{35}+\frac{21\!\cdots\!96}{94\!\cdots\!31}a^{34}+\frac{24\!\cdots\!02}{94\!\cdots\!31}a^{33}+\frac{23\!\cdots\!21}{94\!\cdots\!31}a^{32}-\frac{14\!\cdots\!61}{94\!\cdots\!31}a^{31}+\frac{61\!\cdots\!76}{94\!\cdots\!31}a^{30}+\frac{22\!\cdots\!09}{94\!\cdots\!31}a^{29}-\frac{16\!\cdots\!25}{94\!\cdots\!31}a^{28}+\frac{45\!\cdots\!50}{94\!\cdots\!31}a^{27}+\frac{43\!\cdots\!96}{94\!\cdots\!31}a^{26}-\frac{34\!\cdots\!58}{94\!\cdots\!31}a^{25}-\frac{32\!\cdots\!90}{94\!\cdots\!31}a^{24}-\frac{11\!\cdots\!95}{94\!\cdots\!31}a^{23}+\frac{22\!\cdots\!99}{94\!\cdots\!31}a^{22}-\frac{29\!\cdots\!48}{94\!\cdots\!31}a^{21}-\frac{39\!\cdots\!62}{94\!\cdots\!31}a^{20}-\frac{35\!\cdots\!30}{94\!\cdots\!31}a^{19}+\frac{69\!\cdots\!30}{37\!\cdots\!81}a^{18}+\frac{19\!\cdots\!22}{94\!\cdots\!31}a^{17}+\frac{39\!\cdots\!14}{94\!\cdots\!31}a^{16}+\frac{21\!\cdots\!92}{94\!\cdots\!31}a^{15}+\frac{35\!\cdots\!23}{94\!\cdots\!31}a^{14}-\frac{20\!\cdots\!24}{94\!\cdots\!31}a^{13}+\frac{53\!\cdots\!81}{94\!\cdots\!31}a^{12}+\frac{38\!\cdots\!85}{94\!\cdots\!31}a^{11}-\frac{26\!\cdots\!46}{94\!\cdots\!31}a^{10}+\frac{51\!\cdots\!12}{11\!\cdots\!89}a^{9}+\frac{39\!\cdots\!97}{94\!\cdots\!31}a^{8}-\frac{18\!\cdots\!83}{94\!\cdots\!31}a^{7}+\frac{18\!\cdots\!27}{94\!\cdots\!31}a^{6}-\frac{34\!\cdots\!22}{94\!\cdots\!31}a^{5}-\frac{19\!\cdots\!37}{94\!\cdots\!31}a^{4}+\frac{29\!\cdots\!19}{94\!\cdots\!31}a^{3}-\frac{41\!\cdots\!45}{94\!\cdots\!31}a^{2}+\frac{32\!\cdots\!09}{94\!\cdots\!31}a-\frac{45\!\cdots\!32}{94\!\cdots\!31}$, $\frac{1}{52\!\cdots\!61}a^{41}-\frac{61\!\cdots\!78}{52\!\cdots\!61}a^{40}+\frac{49\!\cdots\!18}{52\!\cdots\!61}a^{39}-\frac{22\!\cdots\!50}{52\!\cdots\!61}a^{38}+\frac{70\!\cdots\!18}{52\!\cdots\!61}a^{37}-\frac{22\!\cdots\!41}{52\!\cdots\!61}a^{36}-\frac{56\!\cdots\!69}{52\!\cdots\!61}a^{35}+\frac{19\!\cdots\!70}{52\!\cdots\!61}a^{34}-\frac{10\!\cdots\!93}{52\!\cdots\!61}a^{33}-\frac{19\!\cdots\!71}{52\!\cdots\!61}a^{32}-\frac{97\!\cdots\!73}{52\!\cdots\!61}a^{31}-\frac{12\!\cdots\!52}{52\!\cdots\!61}a^{30}+\frac{13\!\cdots\!75}{52\!\cdots\!61}a^{29}+\frac{22\!\cdots\!53}{52\!\cdots\!61}a^{28}-\frac{19\!\cdots\!77}{66\!\cdots\!59}a^{27}-\frac{21\!\cdots\!39}{52\!\cdots\!61}a^{26}-\frac{23\!\cdots\!64}{52\!\cdots\!61}a^{25}-\frac{24\!\cdots\!66}{52\!\cdots\!61}a^{24}-\frac{21\!\cdots\!09}{52\!\cdots\!61}a^{23}+\frac{12\!\cdots\!13}{52\!\cdots\!61}a^{22}-\frac{20\!\cdots\!18}{52\!\cdots\!61}a^{21}+\frac{83\!\cdots\!95}{52\!\cdots\!61}a^{20}+\frac{12\!\cdots\!46}{52\!\cdots\!61}a^{19}+\frac{26\!\cdots\!48}{52\!\cdots\!61}a^{18}-\frac{10\!\cdots\!74}{52\!\cdots\!61}a^{17}+\frac{86\!\cdots\!33}{52\!\cdots\!61}a^{16}+\frac{89\!\cdots\!20}{20\!\cdots\!11}a^{15}+\frac{23\!\cdots\!52}{52\!\cdots\!61}a^{14}+\frac{52\!\cdots\!62}{52\!\cdots\!61}a^{13}+\frac{24\!\cdots\!48}{52\!\cdots\!61}a^{12}+\frac{10\!\cdots\!56}{52\!\cdots\!61}a^{11}+\frac{17\!\cdots\!30}{52\!\cdots\!61}a^{10}-\frac{96\!\cdots\!20}{52\!\cdots\!61}a^{9}+\frac{19\!\cdots\!94}{52\!\cdots\!61}a^{8}-\frac{14\!\cdots\!54}{52\!\cdots\!61}a^{7}-\frac{45\!\cdots\!16}{52\!\cdots\!61}a^{6}-\frac{41\!\cdots\!89}{52\!\cdots\!61}a^{5}-\frac{39\!\cdots\!40}{52\!\cdots\!61}a^{4}-\frac{12\!\cdots\!25}{52\!\cdots\!61}a^{3}+\frac{25\!\cdots\!48}{52\!\cdots\!61}a^{2}+\frac{22\!\cdots\!29}{52\!\cdots\!61}a+\frac{25\!\cdots\!25}{52\!\cdots\!61}$ 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:  $20$
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 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947)
 
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 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947, 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 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947);
 
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 - 11*x^41 + 31*x^40 + 15*x^39 + 149*x^38 - 2171*x^37 + 5622*x^36 - 5813*x^35 + 17703*x^34 - 77949*x^33 + 215171*x^32 - 500434*x^31 + 1078714*x^30 - 3010069*x^29 + 7913255*x^28 - 11776682*x^27 + 16679313*x^26 - 87643658*x^25 + 355007978*x^24 - 761687128*x^23 + 908129906*x^22 - 907038305*x^21 + 2012009714*x^20 - 6104907919*x^19 + 18723809190*x^18 - 48743446573*x^17 + 87856746678*x^16 - 105007899193*x^15 + 97597592982*x^14 - 99989052476*x^13 + 101465465536*x^12 - 67260548547*x^11 + 47339909902*x^10 - 82545626769*x^9 + 101683114718*x^8 - 83757841269*x^7 + 103258810875*x^6 - 130872100909*x^5 + 93586325459*x^4 - 47886183228*x^3 + 41077962943*x^2 - 30874848118*x + 9042834947);
 
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 $C_{42}$

Intermediate fields

\(\Q(\sqrt{-43}) \), \(\Q(\zeta_{7})^+\), 6.0.190896307.1, 7.7.6321363049.1, 14.0.1718264124282290785243.1, 21.21.171318696215827426793735775028238670573001.1

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 $42$ $42$ $42$ R $21^{2}$ ${\href{/padicField/13.7.0.1}{7} }^{6}$ $21^{2}$ $42$ $21^{2}$ ${\href{/padicField/29.14.0.1}{14} }^{3}$ $21^{2}$ ${\href{/padicField/37.6.0.1}{6} }^{7}$ ${\href{/padicField/41.7.0.1}{7} }^{6}$ R $21^{2}$ $21^{2}$ $21^{2}$

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 7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 18 x^{5} + 117 x^{4} + 338 x^{3} + 477 x^{2} + 792 x + 1210$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
\(43\) Copy content Toggle raw display 43.14.13.11$x^{14} + 43$$14$$1$$13$$C_{14}$$[\ ]_{14}$
43.14.13.11$x^{14} + 43$$14$$1$$13$$C_{14}$$[\ ]_{14}$
43.14.13.11$x^{14} + 43$$14$$1$$13$$C_{14}$$[\ ]_{14}$