Normalized defining polynomial
\( x^{36} + 3 x^{34} - x^{33} + 9 x^{32} - 6 x^{31} + 28 x^{30} - 27 x^{29} + 90 x^{28} - 109 x^{27} + 297 x^{26} - 417 x^{25} + 1000 x^{24} + 1845 x^{23} + 3417 x^{22} + 4535 x^{21} + 8406 x^{20} + \cdots + 1 \)
Invariants
Degree: | $36$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 18]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(459146050215773460843525344476713987772454059613596693862733\) \(\medspace = 3^{48}\cdot 13^{33}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(45.42\) | 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: | $3^{4/3}13^{11/12}\approx 45.423062587011394$ | ||
Ramified primes: | \(3\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{13}) \) | ||
$\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(117=3^{2}\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{117}(1,·)$, $\chi_{117}(4,·)$, $\chi_{117}(7,·)$, $\chi_{117}(10,·)$, $\chi_{117}(16,·)$, $\chi_{117}(19,·)$, $\chi_{117}(22,·)$, $\chi_{117}(25,·)$, $\chi_{117}(28,·)$, $\chi_{117}(31,·)$, $\chi_{117}(34,·)$, $\chi_{117}(37,·)$, $\chi_{117}(40,·)$, $\chi_{117}(43,·)$, $\chi_{117}(46,·)$, $\chi_{117}(49,·)$, $\chi_{117}(55,·)$, $\chi_{117}(58,·)$, $\chi_{117}(61,·)$, $\chi_{117}(64,·)$, $\chi_{117}(67,·)$, $\chi_{117}(70,·)$, $\chi_{117}(73,·)$, $\chi_{117}(76,·)$, $\chi_{117}(79,·)$, $\chi_{117}(82,·)$, $\chi_{117}(85,·)$, $\chi_{117}(88,·)$, $\chi_{117}(94,·)$, $\chi_{117}(97,·)$, $\chi_{117}(100,·)$, $\chi_{117}(103,·)$, $\chi_{117}(106,·)$, $\chi_{117}(109,·)$, $\chi_{117}(112,·)$, $\chi_{117}(115,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | unavailable$^{131072}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{405821287}a^{25}+\frac{179067424}{405821287}a^{24}+\frac{3652395}{405821287}a^{23}+\frac{131380984}{405821287}a^{22}-\frac{168110239}{405821287}a^{21}-\frac{15330730}{405821287}a^{20}+\frac{175930873}{405821287}a^{19}+\frac{122118049}{405821287}a^{18}+\frac{137302062}{405821287}a^{17}+\frac{190423274}{405821287}a^{16}-\frac{116033150}{405821287}a^{15}+\frac{28146473}{405821287}a^{14}-\frac{132701437}{405821287}a^{13}+\frac{100237981}{405821287}a^{12}+\frac{20436748}{405821287}a^{11}-\frac{132703546}{405821287}a^{10}-\frac{38927737}{405821287}a^{9}-\frac{12726099}{405821287}a^{8}+\frac{15920335}{405821287}a^{7}+\frac{749440}{405821287}a^{6}+\frac{60487104}{405821287}a^{5}-\frac{13672015}{405821287}a^{4}+\frac{180711872}{405821287}a^{3}-\frac{101503149}{405821287}a^{2}+\frac{149986344}{405821287}a-\frac{79400032}{405821287}$, $\frac{1}{405821287}a^{26}-\frac{100234588}{405821287}a^{13}-\frac{145640894}{405821287}$, $\frac{1}{405821287}a^{27}-\frac{100234588}{405821287}a^{14}-\frac{145640894}{405821287}a$, $\frac{1}{405821287}a^{28}-\frac{100234588}{405821287}a^{15}-\frac{145640894}{405821287}a^{2}$, $\frac{1}{405821287}a^{29}-\frac{100234588}{405821287}a^{16}-\frac{145640894}{405821287}a^{3}$, $\frac{1}{405821287}a^{30}-\frac{100234588}{405821287}a^{17}-\frac{145640894}{405821287}a^{4}$, $\frac{1}{405821287}a^{31}-\frac{100234588}{405821287}a^{18}-\frac{145640894}{405821287}a^{5}$, $\frac{1}{405821287}a^{32}-\frac{100234588}{405821287}a^{19}-\frac{145640894}{405821287}a^{6}$, $\frac{1}{405821287}a^{33}-\frac{100234588}{405821287}a^{20}-\frac{145640894}{405821287}a^{7}$, $\frac{1}{405821287}a^{34}-\frac{100234588}{405821287}a^{21}-\frac{145640894}{405821287}a^{8}$, $\frac{1}{405821287}a^{35}-\frac{100234588}{405821287}a^{22}-\frac{145640894}{405821287}a^{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{3}\times C_{3}\times C_{9}\times C_{9}$, which has order $729$ (assuming GRH)
Unit group
Rank: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{10251}{405821287} a^{29} + \frac{34737096}{405821287} a^{16} - \frac{9688000409}{405821287} a^{3} \) (order $26$) | 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: | $\frac{29753}{405821287}a^{30}+\frac{3417}{405821287}a^{28}+\frac{100941399}{405821287}a^{17}+\frac{11579032}{405821287}a^{15}-\frac{27895485399}{405821287}a^{4}-\frac{3364607232}{405821287}a^{2}$, $\frac{29753}{405821287}a^{30}+\frac{100941399}{405821287}a^{17}-\frac{27895485399}{405821287}a^{4}-1$, $\frac{50998407}{405821287}a^{34}-\frac{17711541}{405821287}a^{33}+\frac{152995221}{405821287}a^{32}-\frac{101996814}{405821287}a^{31}+\frac{475985132}{405821287}a^{30}-\frac{458985663}{405821287}a^{29}+\frac{1529952210}{405821287}a^{28}-\frac{1852942121}{405821287}a^{27}+\frac{5048842293}{405821287}a^{26}-\frac{7088778573}{405821287}a^{25}+\frac{16999469000}{405821287}a^{24}-\frac{26315176761}{405821287}a^{23}+\frac{58087185573}{405821287}a^{22}+\frac{77092591915}{405821287}a^{21}+\frac{140481482616}{405821287}a^{20}+\frac{173190590172}{405821287}a^{19}+\frac{351600017327}{405821287}a^{18}+\frac{376674234102}{405821287}a^{17}+\frac{881609461809}{405821287}a^{16}+\frac{778422684979}{405821287}a^{15}+\frac{2268154151325}{405821287}a^{14}+\frac{1453658593128}{405821287}a^{13}+\frac{6026039768996}{405821287}a^{12}+\frac{2092821628059}{405821287}a^{11}+\frac{16624466021955}{405821287}a^{10}+\frac{252425115181}{405821287}a^{9}+\frac{87666261633}{405821287}a^{8}+\frac{696383203038}{405821287}a^{7}+\frac{10573669718}{405821287}a^{6}+\frac{3671885304}{405821287}a^{5}+\frac{1274960175}{405821287}a^{4}+\frac{441986194}{405821287}a^{3}+\frac{152995221}{405821287}a^{2}+\frac{50998407}{405821287}a+\frac{422820756}{405821287}$, $\frac{5903847}{405821287}a^{35}-\frac{48948033}{405821287}a^{34}+\frac{17711541}{405821287}a^{33}-\frac{152995221}{405821287}a^{32}+\frac{101996814}{405821287}a^{31}-\frac{475955379}{405821287}a^{30}+\frac{458995914}{405821287}a^{29}-\frac{1529948793}{405821287}a^{28}+\frac{1852943121}{405821287}a^{27}-\frac{5048842293}{405821287}a^{26}+\frac{7088778573}{405821287}a^{25}-\frac{16999469000}{405821287}a^{24}+\frac{26315176761}{405821287}a^{23}-\frac{38055435285}{405821287}a^{22}-\frac{70135675686}{405821287}a^{21}-\frac{140481482616}{405821287}a^{20}-\frac{173190590172}{405821287}a^{19}-\frac{351600017327}{405821287}a^{18}-\frac{376573292703}{405821287}a^{17}-\frac{881574724713}{405821287}a^{16}-\frac{778411105947}{405821287}a^{15}-\frac{2268150881436}{405821287}a^{14}-\frac{1453658593128}{405821287}a^{13}-\frac{6026039768996}{405821287}a^{12}-\frac{2092821628059}{405821287}a^{11}-\frac{16624466021955}{405821287}a^{10}-\frac{5773616408331}{405821287}a^{9}-\frac{2005155874845}{405821287}a^{8}-\frac{696383203038}{405821287}a^{7}-\frac{10573669718}{405821287}a^{6}-\frac{3671885304}{405821287}a^{5}-\frac{29170445574}{405821287}a^{4}-\frac{10129986603}{405821287}a^{3}-\frac{3517602453}{405821287}a^{2}-\frac{1219514235}{405821287}a-\frac{422820756}{405821287}$, $\frac{140940252}{405821287}a^{35}-\frac{48948033}{405821287}a^{34}+\frac{439820225}{405821287}a^{33}-\frac{293688198}{405821287}a^{32}+\frac{1370459082}{405821287}a^{31}-\frac{1321596891}{405821287}a^{30}+\frac{4405312719}{405821287}a^{29}-\frac{5335335597}{405821287}a^{28}+\frac{14537565801}{405821287}a^{27}-\frac{20411329761}{405821287}a^{26}+\frac{48948033000}{405821287}a^{25}-\frac{75771554084}{405821287}a^{24}+\frac{167255428761}{405821287}a^{23}+\frac{201947579367}{405821287}a^{22}+\frac{411457165398}{405821287}a^{21}+\frac{496266506406}{405821287}a^{20}+\frac{1012392166539}{405821287}a^{19}+\frac{1084299270049}{405821287}a^{18}+\frac{2538493939413}{405821287}a^{17}+\frac{2241344642007}{405821287}a^{16}+\frac{6530891303025}{405821287}a^{15}+\frac{4185644197896}{405821287}a^{14}+\frac{17351294529972}{405821287}a^{13}+\frac{6026041290663}{405821287}a^{12}+\frac{47868242661909}{405821287}a^{11}+\frac{726829342017}{405821287}a^{10}+\frac{5773616299331}{405821287}a^{9}+\frac{87665927103}{405821287}a^{8}+\frac{696382830585}{405821287}a^{7}+\frac{10572775128}{405821287}a^{6}+\frac{83992951440}{405821287}a^{5}+\frac{1272648858}{405821287}a^{4}+\frac{10128532706}{405821287}a^{3}+\frac{146844099}{405821287}a^{2}+\frac{48948033}{405821287}a+1$, $\frac{2417}{405821287}a^{28}+\frac{8309143}{405821287}a^{15}-\frac{1790270117}{405821287}a^{2}$, $\frac{1091027}{405821287}a^{33}+\frac{3701864032}{405821287}a^{20}-\frac{1020274912667}{405821287}a^{7}$, $\frac{5903847}{405821287}a^{35}-\frac{50998407}{405821287}a^{34}+\frac{18337771}{405821287}a^{33}-\frac{152995221}{405821287}a^{32}+\frac{102082656}{405821287}a^{31}-\frac{475955379}{405821287}a^{30}+\frac{458985663}{405821287}a^{29}-\frac{1529952210}{405821287}a^{28}+\frac{1852942121}{405821287}a^{27}-\frac{5048842293}{405821287}a^{26}+\frac{7088778573}{405821287}a^{25}-\frac{16999469000}{405821287}a^{24}+\frac{26315176761}{405821287}a^{23}-\frac{38055435285}{405821287}a^{22}-\frac{77092591915}{405821287}a^{21}-\frac{138356673983}{405821287}a^{20}-\frac{173190590172}{405821287}a^{19}-\frac{351308772162}{405821287}a^{18}-\frac{376573292703}{405821287}a^{17}-\frac{881609461809}{405821287}a^{16}-\frac{778422684979}{405821287}a^{15}-\frac{2268154151325}{405821287}a^{14}-\frac{1453658593128}{405821287}a^{13}-\frac{6026039768996}{405821287}a^{12}-\frac{2092821628059}{405821287}a^{11}-\frac{16624466021955}{405821287}a^{10}-\frac{5773616408331}{405821287}a^{9}-\frac{87666261633}{405821287}a^{8}-\frac{1281998508132}{405821287}a^{7}-\frac{10573669718}{405821287}a^{6}-\frac{83993734269}{405821287}a^{5}-\frac{29170445574}{405821287}a^{4}-\frac{441986194}{405821287}a^{3}-\frac{152995221}{405821287}a^{2}-\frac{50998407}{405821287}a-\frac{16999469}{405821287}$, $\frac{54851880}{405821287}a^{35}+\frac{712072}{405821287}a^{34}+\frac{147556171}{405821287}a^{33}-\frac{48700758}{405821287}a^{32}+\frac{440562050}{405821287}a^{31}-\frac{293658445}{405821287}a^{30}+\frac{1370544924}{405821287}a^{29}-\frac{1321596891}{405821287}a^{28}+\frac{4405322970}{405821287}a^{27}-\frac{5335336597}{405821287}a^{26}+\frac{14537564801}{405821287}a^{25}-\frac{20411329761}{405821287}a^{24}+\frac{48948033000}{405821287}a^{23}+\frac{110340871173}{405821287}a^{22}+\frac{169671482559}{405821287}a^{21}+\frac{224395383453}{405821287}a^{20}+\frac{412296163797}{405821287}a^{19}+\frac{498783501603}{405821287}a^{18}+\frac{1012493107938}{405821287}a^{17}+\frac{1084590515214}{405821287}a^{16}+\frac{2538493939413}{405821287}a^{15}+\frac{2241379379103}{405821287}a^{14}+\frac{6530888033136}{405821287}a^{13}+\frac{4185640928007}{405821287}a^{12}+\frac{17351294529972}{405821287}a^{11}+\frac{6026041290663}{405821287}a^{10}-\frac{3428369194202}{405821287}a^{9}+\frac{60892187958}{405821287}a^{8}-\frac{413512147878}{405821287}a^{7}-\frac{143611619383}{405821287}a^{6}+\frac{2550191127}{405821287}a^{5}-\frac{17322710271}{405821287}a^{4}+\frac{3671102475}{405821287}a^{3}+\frac{1272648858}{405821287}a^{2}+\frac{34711010}{405821287}a+\frac{503717353}{405821287}$, $\frac{5903847}{405821287}a^{35}+\frac{2050374}{405821287}a^{34}+\frac{5834}{405821287}a^{28}+\frac{20031750288}{405821287}a^{22}+\frac{6956916229}{405821287}a^{21}+\frac{19888175}{405821287}a^{15}-\frac{5521191293150}{405821287}a^{9}-\frac{1917489613212}{405821287}a^{8}-\frac{5154877349}{405821287}a^{2}$, $\frac{3853473}{405821287}a^{34}-\frac{10251}{405821287}a^{29}+\frac{13074834059}{405821287}a^{21}-\frac{34737096}{405821287}a^{16}-\frac{3603701679938}{405821287}a^{8}+\frac{9688000409}{405821287}a^{3}+1$, $\frac{69088903}{405821287}a^{35}+\frac{48948033}{405821287}a^{34}+\frac{207266709}{405821287}a^{33}+\frac{77755196}{405821287}a^{32}+\frac{572852094}{405821287}a^{31}+\frac{25998879}{405821287}a^{30}+\frac{1640801086}{405821287}a^{29}-\frac{494855457}{405821287}a^{28}+\frac{4896404379}{405821287}a^{27}-\frac{3125367457}{405821287}a^{26}+\frac{15184070011}{405821287}a^{25}-\frac{14272507750}{405821287}a^{24}+\frac{48677573239}{405821287}a^{23}+\frac{176417059035}{405821287}a^{22}+\frac{326385902436}{405821287}a^{21}+\frac{480573603866}{405821287}a^{20}+\frac{802740648273}{405821287}a^{19}+\frac{1115334909162}{405821287}a^{18}+\frac{1927648340953}{405821287}a^{17}+\frac{2543264079213}{405821287}a^{16}+\frac{4667610113697}{405821287}a^{15}+\frac{5702143896686}{405821287}a^{14}+\frac{11459566261878}{405821287}a^{13}+\frac{12438826615615}{405821287}a^{12}+\frac{28676551619059}{405821287}a^{11}+\frac{25856898467205}{405821287}a^{10}+\frac{8980006427331}{405821287}a^{9}+\frac{3118723219595}{405821287}a^{8}+\frac{1083120814788}{405821287}a^{7}+\frac{376163231454}{405821287}a^{6}+\frac{130639224769}{405821287}a^{5}+\frac{45368879574}{405821287}a^{4}+\frac{15754442853}{405821287}a^{3}+\frac{5467413953}{405821287}a^{2}+\frac{1894448985}{405821287}a+\frac{241977719}{405821287}$, $\frac{5903847}{405821287}a^{35}-\frac{712072}{405821287}a^{34}+\frac{712072}{405821287}a^{33}+\frac{20031750288}{405821287}a^{22}-\frac{2416053798}{405821287}a^{21}+\frac{2416053798}{405821287}a^{20}-\frac{5521191293150}{405821287}a^{9}+\frac{665937154059}{405821287}a^{8}-\frac{665937154059}{405821287}a^{7}$, $\frac{140940252}{405821287}a^{35}+\frac{46897659}{405821287}a^{34}+\frac{408583733}{405821287}a^{33}-\frac{6151122}{405821287}a^{32}+\frac{1178853540}{405821287}a^{31}-\frac{427037099}{405821287}a^{30}+\frac{3542711742}{405821287}a^{29}-\frac{2459964837}{405821287}a^{28}+\frac{11055172325}{405821287}a^{27}-\frac{10922606253}{405821287}a^{26}+\frac{35625481812}{405821287}a^{25}-\frac{43822990084}{405821287}a^{24}+\frac{117799051438}{405821287}a^{23}+\frac{311115822555}{405821287}a^{22}+\frac{556343903138}{405821287}a^{21}+\frac{767242189188}{405821287}a^{20}+\frac{1337884136571}{405821287}a^{19}+\frac{1745382664426}{405821287}a^{18}+\frac{3246410220525}{405821287}a^{17}+\frac{3898263856707}{405821287}a^{16}+\frac{7993847997149}{405821287}a^{15}+\frac{8448381349596}{405821287}a^{14}+\frac{20083280134740}{405821287}a^{13}+\frac{17351296051639}{405821287}a^{12}+\frac{51801462324513}{405821287}a^{11}+\frac{31970605981971}{405821287}a^{10}+\frac{6248020526167}{405821287}a^{9}+\frac{252424671651}{405821287}a^{8}+\frac{87665554650}{405821287}a^{7}+\frac{30444781936}{405821287}a^{6}+\frac{10571992299}{405821287}a^{5}+\frac{3668791158}{405821287}a^{4}+\frac{1271194961}{405821287}a^{3}+\frac{434381175}{405821287}a^{2}+\frac{144793725}{405821287}a+\frac{31948564}{405821287}$, $\frac{2050374}{405821287}a^{34}+\frac{712072}{405821287}a^{33}-\frac{2417}{405821287}a^{27}+\frac{6956916229}{405821287}a^{21}+\frac{2416053798}{405821287}a^{20}-\frac{8309143}{405821287}a^{14}-\frac{1917489613212}{405821287}a^{8}-\frac{665937154059}{405821287}a^{7}+\frac{2196091404}{405821287}a$, $\frac{224978250}{405821287}a^{35}-\frac{77043124}{405821287}a^{34}+\frac{674934750}{405821287}a^{33}-\frac{449956500}{405821287}a^{32}+\frac{2099797000}{405821287}a^{31}-\frac{2024804250}{405821287}a^{30}+\frac{6749347500}{405821287}a^{29}-\frac{8174209750}{405821287}a^{28}+\frac{22272845750}{405821287}a^{27}-\frac{31271976750}{405821287}a^{26}+\frac{74992750000}{405821287}a^{25}-\frac{116088778834}{405821287}a^{24}+\frac{256250226750}{405821287}a^{23}+\frac{340092121250}{405821287}a^{22}+\frac{623432140271}{405821287}a^{21}+\frac{764026137000}{405821287}a^{20}+\frac{1551075048250}{405821287}a^{19}+\frac{1661689354500}{405821287}a^{18}+\frac{3889199007750}{405821287}a^{17}+\frac{3433993015250}{405821287}a^{16}+\frac{10005907668750}{405821287}a^{15}+\frac{6412776768111}{405821287}a^{14}+\frac{26583729991000}{405821287}a^{13}+\frac{9232432445250}{405821287}a^{12}+\frac{73338403126381}{405821287}a^{11}+\frac{1113567344750}{405821287}a^{10}+\frac{386737611750}{405821287}a^{9}+\frac{2051801628462}{405821287}a^{8}+\frac{46645490500}{405821287}a^{7}+\frac{16198434000}{405821287}a^{6}+\frac{5624456250}{405821287}a^{5}+\frac{1949811500}{405821287}a^{4}+\frac{674934750}{405821287}a^{3}+\frac{224978250}{405821287}a^{2}+\frac{1243508578}{405821287}a$, $\frac{85842}{405821287}a^{31}+\frac{3417}{405821287}a^{29}+\frac{1000}{405821287}a^{27}+\frac{291245165}{405821287}a^{18}+\frac{11579032}{405821287}a^{16}+\frac{3269889}{405821287}a^{14}-\frac{80321848965}{405821287}a^{5}-\frac{3364607232}{405821287}a^{3}-\frac{1168515828}{405821287}a$ (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: | \( 20980577392492.816 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 20980577392492.816 \cdot 729}{26\cdot\sqrt{459146050215773460843525344476713987772454059613596693862733}}\cr\approx \mathstrut & 0.202224523334538 \end{aligned}\] (assuming GRH)
Galois group
$C_3\times C_{12}$ (as 36T3):
An abelian group of order 36 |
The 36 conjugacy class representatives for $C_3\times C_{12}$ |
Character table for $C_3\times C_{12}$ is not computed |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.12.0.1}{12} }^{3}$ | R | ${\href{/padicField/5.12.0.1}{12} }^{3}$ | ${\href{/padicField/7.12.0.1}{12} }^{3}$ | ${\href{/padicField/11.12.0.1}{12} }^{3}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.12.0.1}{12} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{12}$ | ${\href{/padicField/31.12.0.1}{12} }^{3}$ | ${\href{/padicField/37.12.0.1}{12} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.12.0.1}{12} }^{3}$ | ${\href{/padicField/53.1.0.1}{1} }^{36}$ | ${\href{/padicField/59.12.0.1}{12} }^{3}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(3\) | 3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ |
3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ | |
3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ | |
3.9.12.1 | $x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$ | $3$ | $3$ | $12$ | $C_3^2$ | $[2]^{3}$ | |
\(13\) | Deg $36$ | $12$ | $3$ | $33$ |