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Properties

Label 40.40.130...936.1
Degree $40$
Signature $[40, 0]$
Discriminant $1.303\times 10^{71}$
Root discriminant \(59.96\)
Ramified primes $2,3,11$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2\times C_{10}$ (as 40T7)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 40*x^38 + 741*x^36 - 8436*x^34 + 66044*x^32 - 376960*x^30 + 1622694*x^28 - 5375528*x^26 + 13860054*x^24 - 27947920*x^22 + 44043506*x^20 - 53927016*x^18 + 50713585*x^16 - 35964944*x^14 + 18713229*x^12 - 6857500*x^10 + 1662386*x^8 - 240976*x^6 + 17560*x^4 - 480*x^2 + 1)
 
gp: K = bnfinit(y^40 - 40*y^38 + 741*y^36 - 8436*y^34 + 66044*y^32 - 376960*y^30 + 1622694*y^28 - 5375528*y^26 + 13860054*y^24 - 27947920*y^22 + 44043506*y^20 - 53927016*y^18 + 50713585*y^16 - 35964944*y^14 + 18713229*y^12 - 6857500*y^10 + 1662386*y^8 - 240976*y^6 + 17560*y^4 - 480*y^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 40*x^38 + 741*x^36 - 8436*x^34 + 66044*x^32 - 376960*x^30 + 1622694*x^28 - 5375528*x^26 + 13860054*x^24 - 27947920*x^22 + 44043506*x^20 - 53927016*x^18 + 50713585*x^16 - 35964944*x^14 + 18713229*x^12 - 6857500*x^10 + 1662386*x^8 - 240976*x^6 + 17560*x^4 - 480*x^2 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 40*x^38 + 741*x^36 - 8436*x^34 + 66044*x^32 - 376960*x^30 + 1622694*x^28 - 5375528*x^26 + 13860054*x^24 - 27947920*x^22 + 44043506*x^20 - 53927016*x^18 + 50713585*x^16 - 35964944*x^14 + 18713229*x^12 - 6857500*x^10 + 1662386*x^8 - 240976*x^6 + 17560*x^4 - 480*x^2 + 1)
 

\( x^{40} - 40 x^{38} + 741 x^{36} - 8436 x^{34} + 66044 x^{32} - 376960 x^{30} + 1622694 x^{28} - 5375528 x^{26} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $40$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[40, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(130305099804548492884220428175380349368393046678311823693003457545895936\) \(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{36}\) 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:  \(59.96\)
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:  $2^{2}3^{1/2}11^{9/10}\approx 59.96171354565991$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $40$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(264=2^{3}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{264}(1,·)$, $\chi_{264}(259,·)$, $\chi_{264}(133,·)$, $\chi_{264}(7,·)$, $\chi_{264}(139,·)$, $\chi_{264}(175,·)$, $\chi_{264}(17,·)$, $\chi_{264}(19,·)$, $\chi_{264}(149,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(155,·)$, $\chi_{264}(29,·)$, $\chi_{264}(161,·)$, $\chi_{264}(37,·)$, $\chi_{264}(41,·)$, $\chi_{264}(43,·)$, $\chi_{264}(173,·)$, $\chi_{264}(47,·)$, $\chi_{264}(49,·)$, $\chi_{264}(179,·)$, $\chi_{264}(181,·)$, $\chi_{264}(59,·)$, $\chi_{264}(151,·)$, $\chi_{264}(191,·)$, $\chi_{264}(65,·)$, $\chi_{264}(197,·)$, $\chi_{264}(71,·)$, $\chi_{264}(203,·)$, $\chi_{264}(79,·)$, $\chi_{264}(211,·)$, $\chi_{264}(169,·)$, $\chi_{264}(101,·)$, $\chi_{264}(97,·)$, $\chi_{264}(229,·)$, $\chi_{264}(233,·)$, $\chi_{264}(157,·)$, $\chi_{264}(119,·)$, $\chi_{264}(251,·)$, $\chi_{264}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $a^{39}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  $39$
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:   $a^{24}-24a^{22}+252a^{20}-1520a^{18}+5814a^{16}-14688a^{14}+24751a^{12}-27444a^{10}+19251a^{8}-7896a^{6}+1611a^{4}-108a^{2}+1$, $a^{28}-28a^{26}+350a^{24}-2576a^{22}+12397a^{20}-40964a^{18}+94961a^{16}-155024a^{14}+176254a^{12}-135784a^{10}+67408a^{8}-19712a^{6}+2849a^{4}-132a^{2}$, $a^{28}-28a^{26}+351a^{24}-2600a^{22}+12649a^{20}-42484a^{18}+100775a^{16}-169712a^{14}+201006a^{12}-163240a^{10}+86713a^{8}-27720a^{6}+4565a^{4}-276a^{2}+2$, $a^{36}-36a^{34}+594a^{32}-5952a^{30}+40454a^{28}-197288a^{26}+712179a^{24}-1934920a^{22}+3983486a^{20}-6206616a^{18}+7253935a^{16}-6248944a^{14}+3855229a^{12}-1627484a^{10}+436601a^{8}-65416a^{6}+4261a^{4}-84a^{2}+1$, $a^{31}-31a^{29}+435a^{27}-3653a^{25}+20449a^{23}-80431a^{21}+228227a^{19}-472037a^{17}+710142a^{15}-766258a^{13}+576433a^{11}-287860a^{9}+87734a^{7}-13998a^{5}+860a^{3}-15a$, $a^{9}-9a^{7}+27a^{5}-30a^{3}+9a$, $a^{3}-3a$, $a^{21}-21a^{19}+189a^{17}-952a^{15}+2940a^{13}-5733a^{11}+7007a^{9}-5148a^{7}+2079a^{5}-385a^{3}+21a$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286825a^{7}+35860a^{5}-2123a^{3}+44a$, $a^{15}-15a^{13}+90a^{11}-275a^{9}+450a^{7}-378a^{5}+140a^{3}-15a-1$, $a^{3}-3a+1$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32319a^{17}+69768a^{15}-104652a^{13}+107406a^{11}-72930a^{9}+30888a^{7}-7371a^{5}+819a^{3}-27a-1$, $a^{33}-33a^{31}+495a^{29}-4466a^{27}+27027a^{25}-115830a^{23}+361790a^{21}-834900a^{19}+1427679a^{17}-1797818a^{15}+1641486a^{13}-1058148a^{11}+461890a^{9}-127908a^{7}+20196a^{5}-1496a^{3}+33a-1$, $a^{31}-31a^{29}+434a^{27}-3627a^{25}+20150a^{23}-78430a^{21}+219604a^{19}-447051a^{17}+660858a^{15}-700911a^{13}+520689a^{11}-260403a^{9}+82368a^{7}-14938a^{5}+1331a^{3}-44a-1$, $a^{33}-34a^{31}+526a^{29}+a^{28}-4901a^{27}-28a^{26}+30681a^{25}+350a^{24}-136304a^{23}-2576a^{22}+442497a^{21}+12397a^{20}-1064899a^{19}-40963a^{18}+1907049a^{17}+94943a^{16}-2528444a^{15}-154889a^{14}+2447049a^{13}+175707a^{12}-1686243a^{11}-134485a^{10}+795223a^{9}+65572a^{8}-241164a^{7}-18214a^{6}+42505a^{5}+2204a^{4}-3646a^{3}-15a^{2}+104a-4$, $a^{34}-34a^{32}-a^{31}+527a^{30}+31a^{29}-4930a^{28}-434a^{27}+31059a^{26}+3627a^{25}-139230a^{24}-20150a^{23}+457470a^{22}+78430a^{21}-1118260a^{20}-219604a^{19}+2042975a^{18}+447051a^{17}-2778446a^{16}-660858a^{15}+2778446a^{14}+700911a^{13}-1998724a^{12}-520689a^{11}+999361a^{10}+260403a^{9}-329450a^{8}-82368a^{7}+65857a^{6}+14938a^{5}-6886a^{4}-1331a^{3}+264a^{2}+44a$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}+a^{30}-319176a^{29}-31a^{28}+1308943a^{27}+433a^{26}-4102110a^{25}-3600a^{24}+9924201a^{23}+19826a^{22}-18597018a^{21}-76153a^{20}+26926515a^{19}+209209a^{18}-29878146a^{17}-414731a^{16}+25040252a^{15}+591074a^{14}-15496248a^{13}-596154a^{12}+6846138a^{11}+412918a^{10}-2051763a^{9}-186748a^{8}+384804a^{7}+50652a^{6}-39285a^{5}-7049a^{4}+1620a^{3}+357a^{2}-1$, $a^{39}-38a^{37}+665a^{35}-7105a^{33}+51798a^{31}-272770a^{29}+1071202a^{27}-3192670a^{25}+7277426a^{23}-12680889a^{21}+16746808a^{19}-16449914a^{17}+11607141a^{15}-5496727a^{13}+1470831a^{11}-60609a^{9}-86316a^{7}+22993a^{5}-1870a^{3}+41a+1$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+a^{30}+45880a^{29}-30a^{28}-232841a^{27}+405a^{26}+878787a^{25}-3250a^{24}-2510820a^{23}+17250a^{22}+5476185a^{21}-63756a^{20}-9126975a^{19}+168245a^{18}+11560835a^{17}-319770a^{16}-10994920a^{15}+436050a^{14}+7696444a^{13}-419900a^{12}-3848222a^{11}+277134a^{10}+1314610a^{9}-119340a^{8}-286825a^{7}+30940a^{6}+35860a^{5}-4200a^{4}-2123a^{3}+225a^{2}+44a-3$, $2a^{38}-76a^{36}+1331a^{34}-14246a^{32}+104190a^{30}-551492a^{28}+2182858a^{26}-6582628a^{24}+15267031a^{22}-27296698a^{20}+37477102a^{18}-39106444a^{16}+30468217a^{14}-17242398a^{12}+6796891a^{10}-1748702a^{8}+263969a^{6}-19430a^{4}+521a^{2}-2$, $a^{30}-30a^{28}+405a^{26}-3250a^{24}+17250a^{22}-63756a^{20}+168245a^{18}-319770a^{16}+436049a^{14}-419886a^{12}+277057a^{10}-119130a^{8}+30646a^{6}-4004a^{4}+176a^{2}$, $a^{38}-38a^{36}+665a^{34}-7106a^{32}+51832a^{30}-273296a^{28}+1076103a^{26}-3223350a^{24}+7413705a^{22}-13123110a^{20}+17809935a^{18}-18349630a^{16}+14115100a^{14}-7904456a^{12}+3105322a^{10}-810084a^{8}+128876a^{6}-10824a^{4}+352a^{2}$, $a^{10}-10a^{8}+35a^{6}-50a^{4}+25a^{2}-2$, $a^{27}-27a^{25}+324a^{23}-2277a^{21}+10395a^{19}-32320a^{17}+69785a^{15}-104771a^{13}+107848a^{11}-73865a^{9}+32010a^{7}-8085a^{5}+1023a^{3}-44a$, $a^{25}-25a^{23}+275a^{21}-1750a^{19}+7125a^{17}-19380a^{15}+35700a^{13}-44200a^{11}+35750a^{9}-17875a^{7}+5005a^{5}-650a^{3}+25a$, $a^{37}-37a^{35}+629a^{33}-6512a^{31}+45880a^{29}-232841a^{27}+878787a^{25}-2510820a^{23}+5476185a^{21}-9126975a^{19}+11560835a^{17}-10994920a^{15}+7696444a^{13}-3848222a^{11}+1314610a^{9}-286824a^{7}+35853a^{5}-2109a^{3}+37a$, $a^{26}-26a^{24}+299a^{22}-2002a^{20}+8645a^{18}-25194a^{16}+50388a^{14}-68952a^{12}+63206a^{10}-37180a^{8}+13013a^{6}-2366a^{4}+169a^{2}-2$, $a^{39}-38a^{37}+665a^{35}-7105a^{33}+51798a^{31}-272770a^{29}+1071202a^{27}-3192670a^{25}+7277426a^{23}-12680889a^{21}+16746808a^{19}-16449914a^{17}+11607141a^{15}-5496727a^{13}+1470831a^{11}-60609a^{9}-86316a^{7}+22993a^{5}-1870a^{3}+40a$, $a^{25}-26a^{23}+299a^{21}-2002a^{19}+8644a^{17}-25176a^{15}+50253a^{13}-68406a^{11}+61919a^{9}-35398a^{7}+11627a^{5}-a^{4}-1826a^{3}+5a^{2}+88a-5$, $a^{35}-35a^{33}+560a^{31}-5425a^{29}+35525a^{27}-166257a^{25}+573300a^{23}-1480050a^{21}+2877875a^{19}-4206125a^{17}+4576264a^{15}-3640210a^{13}+2057510a^{11}-791351a^{9}+193809a^{7}-27159a^{5}+1815a^{3}-44a-1$, $a^{39}-39a^{37}+702a^{35}-7735a^{33}+58344a^{31}-319176a^{29}+1308944a^{27}-4102137a^{25}+9924525a^{23}-18599295a^{21}+26936910a^{19}-29910465a^{17}+25110020a^{15}-15600900a^{13}+6953544a^{11}-2124694a^{9}+415701a^{7}-46684a^{5}+2475a^{3}-44a+1$, $a^{34}-34a^{32}+527a^{30}-4930a^{28}+31059a^{26}-139230a^{24}+457470a^{22}-1118260a^{20}+2042975a^{18}-2778446a^{16}+2778446a^{14}-1998724a^{12}+999362a^{10}-329460a^{8}+65892a^{6}-a^{5}-6936a^{4}+5a^{3}+289a^{2}-5a-2$, $a^{37}-a^{36}-37a^{35}+36a^{34}+629a^{33}-594a^{32}-6512a^{31}+5952a^{30}+45879a^{29}-40454a^{28}-232812a^{27}+197287a^{26}+878410a^{25}-712153a^{24}-2507920a^{23}+1934621a^{22}+5461511a^{21}-3981484a^{20}-9075616a^{19}+6197972a^{18}+11433554a^{17}-7228759a^{16}-10770111a^{15}+6198691a^{14}+7415419a^{13}-3786823a^{12}-3604590a^{11}+1565565a^{10}+1173337a^{9}-401203a^{8}-235103a^{7}+53789a^{6}+24926a^{5}-2435a^{4}-968a^{3}-4a^{2}-1$, $a^{37}+a^{36}-37a^{35}-36a^{34}+629a^{33}+594a^{32}-6512a^{31}-5952a^{30}+45879a^{29}+40454a^{28}-232812a^{27}-197287a^{26}+878410a^{25}+712153a^{24}-2507920a^{23}-1934621a^{22}+5461511a^{21}+3981484a^{20}-9075616a^{19}-6197972a^{18}+11433554a^{17}+7228759a^{16}-10770111a^{15}-6198691a^{14}+7415419a^{13}+3786823a^{12}-3604590a^{11}-1565565a^{10}+1173337a^{9}+401203a^{8}-235103a^{7}-53789a^{6}+24926a^{5}+2435a^{4}-968a^{3}+4a^{2}+1$, $2a^{38}-76a^{36}+1331a^{34}-14246a^{32}+104190a^{30}-551492a^{28}+2182858a^{26}-6582628a^{24}+a^{23}+15267031a^{22}-24a^{21}-27296698a^{20}+251a^{19}+37477102a^{18}-1500a^{17}-39106444a^{16}+5644a^{15}+30468217a^{14}-13888a^{13}-17242398a^{12}+22477a^{11}+6796891a^{10}-23452a^{9}-1748702a^{8}+15015a^{7}+263969a^{6}-5368a^{5}-19430a^{4}+891a^{3}+521a^{2}-44a-2$, $a^{38}+a^{37}-38a^{36}-37a^{35}+665a^{34}+629a^{33}-7106a^{32}-6512a^{31}+51831a^{30}+45880a^{29}-273265a^{28}-232841a^{27}+1075669a^{26}+878786a^{25}-3219724a^{24}-2510795a^{23}+7393580a^{22}+5475909a^{21}-13044955a^{20}-9125203a^{19}+17592082a^{18}+11553502a^{17}-17909723a^{16}-10974436a^{15}+13473774a^{14}+7657140a^{13}-7239911a^{12}-3796573a^{11}+2630575a^{10}+1269202a^{9}-588212a^{8}-261459a^{7}+67038a^{6}+27731a^{5}-2300a^{4}-925a^{3}+16a^{2}+4a-1$, $a+1$, $a^{13}-13a^{11}+65a^{9}-156a^{7}+182a^{5}-91a^{3}+13a-1$, $a^{39}-38a^{37}+666a^{35}-7141a^{33}+52392a^{31}-278722a^{29}+1111656a^{27}-3389958a^{25}+7989605a^{23}-14615809a^{21}+20730294a^{19}-22656530a^{17}+18861076a^{15}-11745671a^{13}+5326060a^{11}-1688093a^{9}+350285a^{7}-42423a^{5}+2390a^{3}-40a-1$ Copy content Toggle raw display (assuming GRH)
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:  \( 73555688392588420000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{40}\cdot(2\pi)^{0}\cdot 73555688392588420000000 \cdot 1}{2\cdot\sqrt{130305099804548492884220428175380349368393046678311823693003457545895936}}\cr\approx \mathstrut & 0.112022533025370 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^40 - 40*x^38 + 741*x^36 - 8436*x^34 + 66044*x^32 - 376960*x^30 + 1622694*x^28 - 5375528*x^26 + 13860054*x^24 - 27947920*x^22 + 44043506*x^20 - 53927016*x^18 + 50713585*x^16 - 35964944*x^14 + 18713229*x^12 - 6857500*x^10 + 1662386*x^8 - 240976*x^6 + 17560*x^4 - 480*x^2 + 1)
 
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^40 - 40*x^38 + 741*x^36 - 8436*x^34 + 66044*x^32 - 376960*x^30 + 1622694*x^28 - 5375528*x^26 + 13860054*x^24 - 27947920*x^22 + 44043506*x^20 - 53927016*x^18 + 50713585*x^16 - 35964944*x^14 + 18713229*x^12 - 6857500*x^10 + 1662386*x^8 - 240976*x^6 + 17560*x^4 - 480*x^2 + 1, 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^40 - 40*x^38 + 741*x^36 - 8436*x^34 + 66044*x^32 - 376960*x^30 + 1622694*x^28 - 5375528*x^26 + 13860054*x^24 - 27947920*x^22 + 44043506*x^20 - 53927016*x^18 + 50713585*x^16 - 35964944*x^14 + 18713229*x^12 - 6857500*x^10 + 1662386*x^8 - 240976*x^6 + 17560*x^4 - 480*x^2 + 1);
 
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^40 - 40*x^38 + 741*x^36 - 8436*x^34 + 66044*x^32 - 376960*x^30 + 1622694*x^28 - 5375528*x^26 + 13860054*x^24 - 27947920*x^22 + 44043506*x^20 - 53927016*x^18 + 50713585*x^16 - 35964944*x^14 + 18713229*x^12 - 6857500*x^10 + 1662386*x^8 - 240976*x^6 + 17560*x^4 - 480*x^2 + 1);
 
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

$C_2^2\times C_{10}$ (as 40T7):

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)
 
An abelian group of order 40
The 40 conjugacy class representatives for $C_2^2\times C_{10}$
Character table for $C_2^2\times C_{10}$ is not computed

Intermediate fields

\(\Q(\sqrt{22}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{66}) \), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{11})\), \(\Q(\sqrt{2}, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\zeta_{11})^+\), 8.8.77720518656.1, 10.10.77265229938688.1, 10.10.1706859170463744.1, \(\Q(\zeta_{33})^+\), 10.10.7024111812608.1, \(\Q(\zeta_{44})^+\), 10.10.53339349076992.1, 10.10.18775450875101184.1, 20.20.352517555563337816067682238201856.2, \(\Q(\zeta_{88})^+\), 20.20.360977976896857923653306611918700544.1, 20.20.2983289065263288625233938941476864.1, 20.20.360977976896857923653306611918700544.2, 20.20.352517555563337816067682238201856.1, \(\Q(\zeta_{132})^+\)

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 R ${\href{/padicField/5.10.0.1}{10} }^{4}$ ${\href{/padicField/7.10.0.1}{10} }^{4}$ R ${\href{/padicField/13.10.0.1}{10} }^{4}$ ${\href{/padicField/17.10.0.1}{10} }^{4}$ ${\href{/padicField/19.10.0.1}{10} }^{4}$ ${\href{/padicField/23.2.0.1}{2} }^{20}$ ${\href{/padicField/29.10.0.1}{10} }^{4}$ ${\href{/padicField/31.10.0.1}{10} }^{4}$ ${\href{/padicField/37.10.0.1}{10} }^{4}$ ${\href{/padicField/41.10.0.1}{10} }^{4}$ ${\href{/padicField/43.2.0.1}{2} }^{20}$ ${\href{/padicField/47.10.0.1}{10} }^{4}$ ${\href{/padicField/53.10.0.1}{10} }^{4}$ ${\href{/padicField/59.10.0.1}{10} }^{4}$

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 $20$$4$$5$$40$
Deg $20$$4$$5$$40$
\(3\) Copy content Toggle raw display 3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
3.20.10.1$x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$$2$$10$$10$20T3$[\ ]_{2}^{10}$
\(11\) Copy content Toggle raw display 11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$
11.20.18.1$x^{20} + 70 x^{19} + 2225 x^{18} + 42420 x^{17} + 539670 x^{16} + 4821684 x^{15} + 31004730 x^{14} + 144683280 x^{13} + 488310165 x^{12} + 1177567510 x^{11} + 1996241675 x^{10} + 2355135790 x^{9} + 1953262935 x^{8} + 1157863560 x^{7} + 500734950 x^{6} + 191763012 x^{5} + 243790230 x^{4} + 806750280 x^{3} + 2014356815 x^{2} + 2999040310 x + 2009802620$$10$$2$$18$20T3$[\ ]_{10}^{2}$