Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^40 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641)
gp: K = bnfinit(y^40 - 33*y^36 + 737*y^32 - 8954*y^28 + 78650*y^24 - 380787*y^20 + 1299056*y^16 - 1686377*y^12 + 1610510*y^8 - 161051*y^4 + 14641, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^40 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^40 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641)
\( x^{40} - 33 x^{36} + 737 x^{32} - 8954 x^{28} + 78650 x^{24} - 380787 x^{20} + 1299056 x^{16} + \cdots + 14641 \)
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: | $[0, 20]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(130305099804548492884220428175380349368393046678311823693003457545895936\)
\(\medspace = 2^{80}\cdot 3^{20}\cdot 11^{36}\)
| 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: |
\(2\), \(3\), \(11\)
| 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$$\chi_{264}(1,·)$, $\chi_{264}(131,·)$, $\chi_{264}(257,·)$, $\chi_{264}(137,·)$, $\chi_{264}(139,·)$, $\chi_{264}(13,·)$, $\chi_{264}(19,·)$, $\chi_{264}(149,·)$, $\chi_{264}(23,·)$, $\chi_{264}(25,·)$, $\chi_{264}(29,·)$, $\chi_{264}(31,·)$, $\chi_{264}(35,·)$, $\chi_{264}(169,·)$, $\chi_{264}(43,·)$, $\chi_{264}(173,·)$, $\chi_{264}(47,·)$, $\chi_{264}(49,·)$, $\chi_{264}(185,·)$, $\chi_{264}(61,·)$, $\chi_{264}(191,·)$, $\chi_{264}(197,·)$, $\chi_{264}(71,·)$, $\chi_{264}(119,·)$, $\chi_{264}(205,·)$, $\chi_{264}(83,·)$, $\chi_{264}(85,·)$, $\chi_{264}(89,·)$, $\chi_{264}(199,·)$, $\chi_{264}(223,·)$, $\chi_{264}(97,·)$, $\chi_{264}(227,·)$, $\chi_{264}(101,·)$, $\chi_{264}(103,·)$, $\chi_{264}(259,·)$, $\chi_{264}(107,·)$, $\chi_{264}(109,·)$, $\chi_{264}(113,·)$, $\chi_{264}(211,·)$, $\chi_{264}(247,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{524288}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11}a^{10}$, $\frac{1}{11}a^{11}$, $\frac{1}{11}a^{12}$, $\frac{1}{11}a^{13}$, $\frac{1}{11}a^{14}$, $\frac{1}{11}a^{15}$, $\frac{1}{11}a^{16}$, $\frac{1}{11}a^{17}$, $\frac{1}{11}a^{18}$, $\frac{1}{11}a^{19}$, $\frac{1}{121}a^{20}$, $\frac{1}{121}a^{21}$, $\frac{1}{121}a^{22}$, $\frac{1}{121}a^{23}$, $\frac{1}{121}a^{24}$, $\frac{1}{121}a^{25}$, $\frac{1}{121}a^{26}$, $\frac{1}{121}a^{27}$, $\frac{1}{13189}a^{28}+\frac{43}{13189}a^{24}+\frac{24}{13189}a^{20}+\frac{53}{1199}a^{16}-\frac{48}{1199}a^{12}-\frac{23}{109}a^{8}-\frac{27}{109}a^{4}+\frac{48}{109}$, $\frac{1}{13189}a^{29}+\frac{43}{13189}a^{25}+\frac{24}{13189}a^{21}+\frac{53}{1199}a^{17}-\frac{48}{1199}a^{13}-\frac{23}{109}a^{9}-\frac{27}{109}a^{5}+\frac{48}{109}a$, $\frac{1}{145079}a^{30}-\frac{6}{13189}a^{26}+\frac{2}{1199}a^{22}-\frac{15}{1199}a^{18}-\frac{4}{109}a^{14}-\frac{23}{1199}a^{10}-\frac{52}{109}a^{6}+\frac{44}{109}a^{2}$, $\frac{1}{145079}a^{31}-\frac{6}{13189}a^{27}+\frac{2}{1199}a^{23}-\frac{15}{1199}a^{19}-\frac{4}{109}a^{15}-\frac{23}{1199}a^{11}-\frac{52}{109}a^{7}+\frac{44}{109}a^{3}$, $\frac{1}{555217333}a^{32}-\frac{1271}{50474303}a^{28}-\frac{17914}{4588573}a^{24}-\frac{193916}{50474303}a^{20}-\frac{34607}{4588573}a^{16}-\frac{28901}{4588573}a^{12}+\frac{174667}{417143}a^{8}+\frac{1499}{417143}a^{4}+\frac{32802}{417143}$, $\frac{1}{555217333}a^{33}-\frac{1271}{50474303}a^{29}-\frac{17914}{4588573}a^{25}-\frac{193916}{50474303}a^{21}-\frac{34607}{4588573}a^{17}-\frac{28901}{4588573}a^{13}+\frac{174667}{417143}a^{9}+\frac{1499}{417143}a^{5}+\frac{32802}{417143}a$, $\frac{1}{555217333}a^{34}+\frac{1327}{555217333}a^{30}+\frac{128241}{50474303}a^{26}+\frac{142860}{50474303}a^{22}+\frac{152916}{4588573}a^{18}+\frac{131833}{4588573}a^{14}-\frac{9029}{417143}a^{10}+\frac{39769}{417143}a^{6}-\frac{127932}{417143}a^{2}$, $\frac{1}{555217333}a^{35}+\frac{1327}{555217333}a^{31}+\frac{128241}{50474303}a^{27}+\frac{142860}{50474303}a^{23}+\frac{152916}{4588573}a^{19}+\frac{131833}{4588573}a^{15}-\frac{9029}{417143}a^{11}+\frac{39769}{417143}a^{7}-\frac{127932}{417143}a^{3}$, $\frac{1}{977195276078659}a^{36}+\frac{15820}{977195276078659}a^{32}+\frac{652247789}{88835934188969}a^{28}-\frac{49834937375}{88835934188969}a^{24}+\frac{326488951231}{88835934188969}a^{20}-\frac{330541501756}{8075994017179}a^{16}-\frac{314178082332}{8075994017179}a^{12}+\frac{26337609990}{734181274289}a^{8}-\frac{181590528712}{734181274289}a^{4}-\frac{326842105055}{734181274289}$, $\frac{1}{977195276078659}a^{37}+\frac{15820}{977195276078659}a^{33}+\frac{652247789}{88835934188969}a^{29}-\frac{49834937375}{88835934188969}a^{25}+\frac{326488951231}{88835934188969}a^{21}-\frac{330541501756}{8075994017179}a^{17}-\frac{314178082332}{8075994017179}a^{13}+\frac{26337609990}{734181274289}a^{9}-\frac{181590528712}{734181274289}a^{5}-\frac{326842105055}{734181274289}a$, $\frac{1}{977195276078659}a^{38}+\frac{15820}{977195276078659}a^{34}+\frac{439117658}{977195276078659}a^{30}-\frac{9421289249}{88835934188969}a^{26}+\frac{178305574769}{88835934188969}a^{22}-\frac{229507381441}{8075994017179}a^{18}-\frac{17811329408}{8075994017179}a^{14}-\frac{289548579916}{8075994017179}a^{10}+\frac{168661088380}{734181274289}a^{6}+\frac{110972416310}{734181274289}a^{2}$, $\frac{1}{977195276078659}a^{39}+\frac{15820}{977195276078659}a^{35}+\frac{439117658}{977195276078659}a^{31}-\frac{9421289249}{88835934188969}a^{27}+\frac{178305574769}{88835934188969}a^{23}-\frac{229507381441}{8075994017179}a^{19}-\frac{17811329408}{8075994017179}a^{15}-\frac{289548579916}{8075994017179}a^{11}+\frac{168661088380}{734181274289}a^{7}+\frac{110972416310}{734181274289}a^{3}$
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
$C_{2728}$, which has order $2728$ (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: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{1720057043}{977195276078659} a^{38} + \frac{5002106944}{88835934188969} a^{34} - \frac{1210666196290}{977195276078659} a^{30} + \frac{1284644679822}{88835934188969} a^{26} - \frac{10905062085446}{88835934188969} a^{22} + \frac{4305094604522}{8075994017179} a^{18} - \frac{13178268444118}{8075994017179} a^{14} + \frac{5995963067436}{8075994017179} a^{10} - \frac{54554228858}{734181274289} a^{6} - \frac{2308120940327}{734181274289} a^{2} \)
(order $12$)
| 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: |
$\frac{2906161311}{977195276078659}a^{38}-\frac{8449091136}{88835934188969}a^{34}+\frac{2044944089010}{977195276078659}a^{30}-\frac{2169572874609}{88835934188969}a^{26}+\frac{18419810778774}{88835934188969}a^{22}-\frac{7271763093018}{8075994017179}a^{18}+\frac{22279564287842}{8075994017179}a^{14}-\frac{10127819931084}{8075994017179}a^{10}+\frac{92147900202}{734181274289}a^{6}+\frac{3021361321624}{734181274289}a^{2}$, $\frac{1119193994}{977195276078659}a^{36}-\frac{3275503944}{88835934188969}a^{32}+\frac{72557158538}{88835934188969}a^{28}-\frac{78129126882}{8075994017179}a^{24}+\frac{7462871523868}{88835934188969}a^{20}-\frac{3142751611009}{8075994017179}a^{16}+\frac{10935886749086}{8075994017179}a^{12}-\frac{1232239114266}{734181274289}a^{8}+\frac{2197002228504}{734181274289}a^{4}-\frac{11503134062}{734181274289}$, $\frac{4535391194}{977195276078659}a^{38}-\frac{13192897792}{88835934188969}a^{34}+\frac{3193093543720}{977195276078659}a^{30}-\frac{308070939340}{8075994017179}a^{26}+\frac{28761754020728}{88835934188969}a^{22}-\frac{11354549940296}{8075994017179}a^{18}+\frac{34694201264119}{8075994017179}a^{14}-\frac{15814161672048}{8075994017179}a^{10}+\frac{143885041544}{734181274289}a^{6}+\frac{4222292632852}{734181274289}a^{2}$, $\frac{17309863072}{977195276078659}a^{38}-\frac{572534813573}{977195276078659}a^{34}+\frac{1163555883552}{88835934188969}a^{30}-\frac{14173730592038}{88835934188969}a^{26}+\frac{124738160280431}{88835934188969}a^{22}-\frac{55226282872466}{8075994017179}a^{18}+\frac{17191357281246}{734181274289}a^{14}-\frac{251418186737936}{8075994017179}a^{10}+\frac{21358487795332}{734181274289}a^{6}-\frac{2135883280058}{734181274289}a^{2}$, $\frac{21332264457}{977195276078659}a^{38}-\frac{703339947146}{977195276078659}a^{34}+\frac{1427448295579}{88835934188969}a^{30}-\frac{17324562809348}{88835934188969}a^{26}+\frac{152060537160824}{88835934188969}a^{22}-\frac{66773233315669}{8075994017179}a^{18}+\frac{20678405212462}{734181274289}a^{14}-\frac{292467165668974}{8075994017179}a^{10}+\frac{25614428399310}{734181274289}a^{6}-\frac{2561426354267}{734181274289}a^{2}-1$, $\frac{6880228172}{977195276078659}a^{38}-\frac{17948030420}{977195276078659}a^{36}-\frac{20008427776}{88835934188969}a^{34}+\frac{589104956875}{977195276078659}a^{32}+\frac{4842664785160}{977195276078659}a^{30}-\frac{1192574936632}{88835934188969}a^{28}-\frac{5138578719288}{88835934188969}a^{26}+\frac{14383865609066}{88835934188969}a^{24}+\frac{43620248341784}{88835934188969}a^{22}-\frac{125464771299826}{88835934188969}a^{20}-\frac{17220378418088}{8075994017179}a^{18}+\frac{54133611891682}{8075994017179}a^{16}+\frac{52713073776472}{8075994017179}a^{14}-\frac{180433865978580}{8075994017179}a^{12}-\frac{23983852269744}{8075994017179}a^{10}+\frac{18864641259970}{734181274289}a^{8}+\frac{218216915432}{734181274289}a^{6}-\frac{15873430412338}{734181274289}a^{4}+\frac{8498302487019}{734181274289}a^{2}-\frac{1233990138630}{734181274289}$, $\frac{66229840223}{977195276078659}a^{38}-\frac{24122533}{998156563921}a^{36}-\frac{2179359759961}{977195276078659}a^{34}+\frac{777129236318}{977195276078659}a^{32}+\frac{48606736961820}{977195276078659}a^{30}-\frac{1575858954426}{88835934188969}a^{28}-\frac{53496281223296}{88835934188969}a^{26}+\frac{19082049254654}{88835934188969}a^{24}+\frac{468526050708253}{88835934188969}a^{22}-\frac{167195740033428}{88835934188969}a^{20}-\frac{204432394048622}{8075994017179}a^{18}+\frac{73031919789342}{8075994017179}a^{16}+\frac{691902314476107}{8075994017179}a^{14}-\frac{247843512639603}{8075994017179}a^{12}-\frac{78014785710488}{734181274289}a^{10}+\frac{28292934715920}{734181274289}a^{8}+\frac{72785783864354}{734181274289}a^{6}-\frac{27354121160758}{734181274289}a^{4}-\frac{16937730591}{17073983123}a^{2}+\frac{34876424475}{17073983123}$, $\frac{5824487}{368613834809}a^{38}-\frac{137157209}{10979722203131}a^{36}-\frac{2106959755}{4054752182899}a^{34}+\frac{403772652504}{977195276078659}a^{32}+\frac{46983312328}{4054752182899}a^{30}-\frac{820437326791}{88835934188969}a^{28}-\frac{51678104048}{368613834809}a^{26}+\frac{9991829795248}{88835934188969}a^{24}+\frac{41136122821}{33510348619}a^{22}-\frac{87876965194267}{88835934188969}a^{20}-\frac{197239964111}{33510348619}a^{18}+\frac{38856388668373}{8075994017179}a^{16}+\frac{668092403153}{33510348619}a^{14}-\frac{132543905526938}{8075994017179}a^{12}-\frac{75328767704}{3046395329}a^{10}+\frac{15822953476326}{734181274289}a^{8}+\frac{73281808060}{3046395329}a^{6}-\frac{13694001570135}{734181274289}a^{4}-\frac{703245455}{3046395329}a^{2}+\frac{758422677989}{734181274289}$, $\frac{637004267}{88835934188969}a^{38}+\frac{2752224943}{977195276078659}a^{36}-\frac{229056738374}{977195276078659}a^{34}-\frac{729174368}{8075994017179}a^{32}+\frac{5092951776893}{977195276078659}a^{30}+\frac{176482983755}{88835934188969}a^{28}-\frac{5560139850927}{88835934188969}a^{26}-\frac{2062607355250}{88835934188969}a^{24}+\frac{48381871444382}{88835934188969}a^{22}+\frac{1589668482337}{8075994017179}a^{20}-\frac{1882590865190}{734181274289}a^{18}-\frac{627568477159}{734181274289}a^{16}+\frac{69081001550945}{8075994017179}a^{14}+\frac{20885558792667}{8075994017179}a^{12}-\frac{78902222117767}{8075994017179}a^{10}-\frac{874052200242}{734181274289}a^{8}+\frac{7250677444391}{734181274289}a^{6}+\frac{87478137461}{734181274289}a^{4}-\frac{15537418117}{8249227801}a^{2}-\frac{66417003877}{734181274289}$, $\frac{82352623770}{977195276078659}a^{39}+\frac{21332264457}{977195276078659}a^{38}+\frac{77859977}{10979722203131}a^{36}-\frac{2709945738851}{977195276078659}a^{35}-\frac{703339947146}{977195276078659}a^{34}-\frac{2559293760}{10979722203131}a^{32}+\frac{60440846381030}{977195276078659}a^{31}+\frac{1427448295579}{88835934188969}a^{30}+\frac{5187423931}{998156563921}a^{28}-\frac{66522015086119}{88835934188969}a^{27}-\frac{17324562809348}{88835934188969}a^{26}-\frac{62735658723}{998156563921}a^{24}+\frac{582610769883369}{88835934188969}a^{23}+\frac{152060537160824}{88835934188969}a^{22}+\frac{549223371586}{998156563921}a^{20}-\frac{254218426568395}{8075994017179}a^{19}-\frac{66773233315669}{8075994017179}a^{18}-\frac{239239274162}{90741505811}a^{16}+\frac{860383775109920}{8075994017179}a^{15}+\frac{20678405212462}{734181274289}a^{14}+\frac{73707114553}{8249227801}a^{12}-\frac{97011803439817}{734181274289}a^{11}-\frac{292467165668974}{8075994017179}a^{10}-\frac{91415662629}{8249227801}a^{8}+\frac{90353167869810}{734181274289}a^{7}+\frac{25614428399310}{734181274289}a^{6}+\frac{91029198008}{8249227801}a^{4}-\frac{905672862398}{734181274289}a^{3}-\frac{2561426354267}{734181274289}a^{2}-\frac{853426595}{8249227801}$, $\frac{1720057043}{977195276078659}a^{38}-\frac{15195805477}{977195276078659}a^{37}+\frac{77859977}{10979722203131}a^{36}-\frac{5002106944}{88835934188969}a^{34}+\frac{500874858347}{977195276078659}a^{33}-\frac{2559293760}{10979722203131}a^{32}+\frac{1210666196290}{977195276078659}a^{30}-\frac{1016091952877}{88835934188969}a^{29}+\frac{5187423931}{998156563921}a^{28}-\frac{1284644679822}{88835934188969}a^{26}+\frac{12321258253816}{88835934188969}a^{25}-\frac{62735658723}{998156563921}a^{24}+\frac{10905062085446}{88835934188969}a^{22}-\frac{107978417994119}{88835934188969}a^{21}+\frac{549223371586}{998156563921}a^{20}-\frac{4305094604522}{8075994017179}a^{18}+\frac{47230358642933}{8075994017179}a^{17}-\frac{239239274162}{90741505811}a^{16}+\frac{13178268444118}{8075994017179}a^{14}-\frac{159548307185913}{8075994017179}a^{13}+\frac{73707114553}{8249227801}a^{12}-\frac{5995963067436}{8075994017179}a^{10}+\frac{17990589059728}{734181274289}a^{9}-\frac{91415662629}{8249227801}a^{8}+\frac{54554228858}{734181274289}a^{6}-\frac{15785952274877}{734181274289}a^{5}+\frac{91029198008}{8249227801}a^{4}+\frac{2308120940327}{734181274289}a^{2}+\frac{167955406071}{734181274289}a-\frac{9102654396}{8249227801}$, $\frac{21332264457}{977195276078659}a^{38}-\frac{624779935}{977195276078659}a^{37}-\frac{77859977}{10979722203131}a^{36}-\frac{703339947146}{977195276078659}a^{34}+\frac{164856640}{8075994017179}a^{33}+\frac{2559293760}{10979722203131}a^{32}+\frac{1427448295579}{88835934188969}a^{30}-\frac{39900458650}{88835934188969}a^{29}-\frac{5187423931}{998156563921}a^{28}-\frac{17324562809348}{88835934188969}a^{26}+\frac{465153706726}{88835934188969}a^{25}+\frac{62735658723}{998156563921}a^{24}+\frac{152060537160824}{88835934188969}a^{22}-\frac{359402930510}{8075994017179}a^{21}-\frac{549223371586}{998156563921}a^{20}-\frac{66773233315669}{8075994017179}a^{18}+\frac{141884897570}{734181274289}a^{17}+\frac{239239274162}{90741505811}a^{16}+\frac{20678405212462}{734181274289}a^{14}-\frac{4840604068235}{8075994017179}a^{13}-\frac{73707114553}{8249227801}a^{12}-\frac{292467165668974}{8075994017179}a^{10}+\frac{197611593660}{734181274289}a^{9}+\frac{91415662629}{8249227801}a^{8}+\frac{25614428399310}{734181274289}a^{6}-\frac{19777645030}{734181274289}a^{5}-\frac{91029198008}{8249227801}a^{4}-\frac{2561426354267}{734181274289}a^{2}-\frac{1967888913840}{734181274289}a+\frac{853426595}{8249227801}$, $\frac{6178101258}{88835934188969}a^{39}-\frac{1720057043}{977195276078659}a^{38}+\frac{77859977}{10979722203131}a^{36}-\frac{2244662647037}{977195276078659}a^{35}+\frac{5002106944}{88835934188969}a^{34}-\frac{2559293760}{10979722203131}a^{32}+\frac{414464824222}{8075994017179}a^{31}-\frac{1210666196290}{977195276078659}a^{30}+\frac{5187423931}{998156563921}a^{28}-\frac{55447488954892}{88835934188969}a^{27}+\frac{1284644679822}{88835934188969}a^{26}-\frac{62735658723}{998156563921}a^{24}+\frac{487411116919002}{88835934188969}a^{23}-\frac{10905062085446}{88835934188969}a^{22}+\frac{549223371586}{998156563921}a^{20}-\frac{215027228187410}{8075994017179}a^{19}+\frac{4305094604522}{8075994017179}a^{18}-\frac{239239274162}{90741505811}a^{16}+\frac{66785998602056}{734181274289}a^{15}-\frac{13178268444118}{8075994017179}a^{14}+\frac{73707114553}{8249227801}a^{12}-\frac{962485654171210}{8075994017179}a^{11}+\frac{5995963067436}{8075994017179}a^{10}-\frac{91415662629}{8249227801}a^{8}+\frac{82868277364764}{734181274289}a^{7}-\frac{54554228858}{734181274289}a^{6}+\frac{91029198008}{8249227801}a^{4}-\frac{8286880940470}{734181274289}a^{3}-\frac{2308120940327}{734181274289}a^{2}-\frac{9102654396}{8249227801}$, $\frac{8001500}{339186142339}a^{38}+\frac{77859977}{10979722203131}a^{37}-\frac{263229130}{339186142339}a^{34}-\frac{2559293760}{10979722203131}a^{33}+\frac{5870391339}{339186142339}a^{30}+\frac{5187423931}{998156563921}a^{29}-\frac{6459287570}{30835103849}a^{26}-\frac{62735658723}{998156563921}a^{25}+\frac{56565636670}{30835103849}a^{22}+\frac{549223371586}{998156563921}a^{21}-\frac{24671408511}{2803191259}a^{18}-\frac{239239274162}{90741505811}a^{17}+\frac{83526805200}{2803191259}a^{14}+\frac{73707114553}{8249227801}a^{13}-\frac{9417914510}{254835569}a^{10}-\frac{91415662629}{8249227801}a^{9}+\frac{8909747528}{254835569}a^{6}+\frac{91029198008}{8249227801}a^{5}-\frac{87922740}{254835569}a^{2}-\frac{9102654396}{8249227801}a-1$, $\frac{21332264457}{977195276078659}a^{38}-\frac{15195805477}{977195276078659}a^{37}-\frac{703339947146}{977195276078659}a^{34}+\frac{500874858347}{977195276078659}a^{33}+\frac{1427448295579}{88835934188969}a^{30}-\frac{1016091952877}{88835934188969}a^{29}-\frac{17324562809348}{88835934188969}a^{26}+\frac{12321258253816}{88835934188969}a^{25}+\frac{152060537160824}{88835934188969}a^{22}-\frac{107978417994119}{88835934188969}a^{21}-\frac{66773233315669}{8075994017179}a^{18}+\frac{47230358642933}{8075994017179}a^{17}+\frac{20678405212462}{734181274289}a^{14}-\frac{159548307185913}{8075994017179}a^{13}-\frac{292467165668974}{8075994017179}a^{10}+\frac{17990589059728}{734181274289}a^{9}+\frac{25614428399310}{734181274289}a^{6}-\frac{15785952274877}{734181274289}a^{5}-\frac{2561426354267}{734181274289}a^{2}+\frac{167955406071}{734181274289}a+1$, $\frac{6178101258}{88835934188969}a^{39}+\frac{8001500}{339186142339}a^{38}-\frac{77859977}{10979722203131}a^{36}-\frac{2244662647037}{977195276078659}a^{35}-\frac{263229130}{339186142339}a^{34}+\frac{2559293760}{10979722203131}a^{32}+\frac{414464824222}{8075994017179}a^{31}+\frac{5870391339}{339186142339}a^{30}-\frac{5187423931}{998156563921}a^{28}-\frac{55447488954892}{88835934188969}a^{27}-\frac{6459287570}{30835103849}a^{26}+\frac{62735658723}{998156563921}a^{24}+\frac{487411116919002}{88835934188969}a^{23}+\frac{56565636670}{30835103849}a^{22}-\frac{549223371586}{998156563921}a^{20}-\frac{215027228187410}{8075994017179}a^{19}-\frac{24671408511}{2803191259}a^{18}+\frac{239239274162}{90741505811}a^{16}+\frac{66785998602056}{734181274289}a^{15}+\frac{83526805200}{2803191259}a^{14}-\frac{73707114553}{8249227801}a^{12}-\frac{962485654171210}{8075994017179}a^{11}-\frac{9417914510}{254835569}a^{10}+\frac{91415662629}{8249227801}a^{8}+\frac{82868277364764}{734181274289}a^{7}+\frac{8909747528}{254835569}a^{6}-\frac{91029198008}{8249227801}a^{4}-\frac{8286880940470}{734181274289}a^{3}-\frac{87922740}{254835569}a^{2}+\frac{853426595}{8249227801}$, $\frac{523552802}{8965094275951}a^{39}-\frac{21332264457}{977195276078659}a^{38}-\frac{17268281622}{8965094275951}a^{35}+\frac{703339947146}{977195276078659}a^{34}+\frac{35051964742}{815008570541}a^{31}-\frac{1427448295579}{88835934188969}a^{30}-\frac{425598300933}{815008570541}a^{27}+\frac{17324562809348}{88835934188969}a^{26}+\frac{3736703957902}{815008570541}a^{23}-\frac{152060537160824}{88835934188969}a^{22}-\frac{1642453252721}{74091688231}a^{19}+\frac{66773233315669}{8075994017179}a^{18}+\frac{508947545341}{6735608021}a^{15}-\frac{20678405212462}{734181274289}a^{14}-\frac{7228491406359}{74091688231}a^{11}+\frac{292467165668974}{8075994017179}a^{10}+\frac{630657675002}{6735608021}a^{7}-\frac{25614428399310}{734181274289}a^{6}-\frac{63065530473}{6735608021}a^{3}+\frac{2561426354267}{734181274289}a^{2}+1$, $\frac{8001500}{339186142339}a^{38}-\frac{25954159590}{977195276078659}a^{37}+\frac{77859977}{10979722203131}a^{36}-\frac{263229130}{339186142339}a^{34}+\frac{853806501195}{977195276078659}a^{33}-\frac{2559293760}{10979722203131}a^{32}+\frac{5870391339}{339186142339}a^{30}-\frac{1730994883395}{88835934188969}a^{29}+\frac{5187423931}{998156563921}a^{28}-\frac{6459287570}{30835103849}a^{26}+\frac{20950639271715}{88835934188969}a^{25}-\frac{62735658723}{998156563921}a^{24}+\frac{56565636670}{30835103849}a^{22}-\frac{183468530316825}{88835934188969}a^{21}+\frac{549223371586}{998156563921}a^{20}-\frac{24671408511}{2803191259}a^{18}+\frac{80020157154440}{8075994017179}a^{17}-\frac{239239274162}{90741505811}a^{16}+\frac{83526805200}{2803191259}a^{14}-\frac{270913966032660}{8075994017179}a^{13}+\frac{73707114553}{8249227801}a^{12}-\frac{9417914510}{254835569}a^{10}+\frac{30546395904045}{734181274289}a^{9}-\frac{91415662629}{8249227801}a^{8}+\frac{8909747528}{254835569}a^{6}-\frac{28660935175199}{734181274289}a^{5}+\frac{91029198008}{8249227801}a^{4}-\frac{87922740}{254835569}a^{2}+\frac{285171697410}{734181274289}a-\frac{853426595}{8249227801}$, $\frac{8001500}{339186142339}a^{38}-\frac{77859977}{10979722203131}a^{36}-\frac{263229130}{339186142339}a^{34}+\frac{2559293760}{10979722203131}a^{32}+\frac{5870391339}{339186142339}a^{30}-\frac{5187423931}{998156563921}a^{28}-\frac{6459287570}{30835103849}a^{26}+\frac{62735658723}{998156563921}a^{24}+\frac{56565636670}{30835103849}a^{22}-\frac{549223371586}{998156563921}a^{20}-\frac{24671408511}{2803191259}a^{18}+\frac{239239274162}{90741505811}a^{16}+\frac{83526805200}{2803191259}a^{14}-\frac{73707114553}{8249227801}a^{12}-\frac{9417914510}{254835569}a^{10}+\frac{91415662629}{8249227801}a^{8}+\frac{8909747528}{254835569}a^{6}-\frac{91029198008}{8249227801}a^{4}-\frac{87922740}{254835569}a^{2}-a+\frac{9102654396}{8249227801}$
| 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: | \( 24039412784024376 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{20}\cdot 24039412784024376 \cdot 2728}{12\cdot\sqrt{130305099804548492884220428175380349368393046678311823693003457545895936}}\cr\approx \mathstrut & 0.139220643032544 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^40 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641)
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 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641, 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 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641);
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 - 33*x^36 + 737*x^32 - 8954*x^28 + 78650*x^24 - 380787*x^20 + 1299056*x^16 - 1686377*x^12 + 1610510*x^8 - 161051*x^4 + 14641);
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 $C_2^2\times C_{10}$ |
Intermediate fields
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 | ${\href{/padicField/13.5.0.1}{5} }^{8}$ | ${\href{/padicField/17.10.0.1}{10} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{4}$ | ${\href{/padicField/23.2.0.1}{2} }^{20}$ | ${\href{/padicField/29.10.0.1}{10} }^{4}$ | ${\href{/padicField/31.10.0.1}{10} }^{4}$ | ${\href{/padicField/37.10.0.1}{10} }^{4}$ | ${\href{/padicField/41.10.0.1}{10} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{20}$ | ${\href{/padicField/47.10.0.1}{10} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{4}$ | ${\href{/padicField/59.10.0.1}{10} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $40$ | $4$ | $10$ | $80$ | |||
|
\(3\)
| 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 | $[\ ]_{2}^{10}$ | |
|
\(11\)
| 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 | $[\ ]_{10}^{2}$ |