Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 1)
gp: K = bnfinit(y^36 + 234*y^32 + 16497*y^28 + 423088*y^24 + 3439800*y^20 + 2847312*y^16 + 761144*y^12 + 73593*y^8 + 1794*y^4 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 1)
\( x^{36} + 234 x^{32} + 16497 x^{28} + 423088 x^{24} + 3439800 x^{20} + 2847312 x^{16} + 761144 x^{12} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
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: |
\(72166039996680932841568261508505246616460162082637588781812678656\)
\(\medspace = 2^{72}\cdot 3^{48}\cdot 7^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(63.33\) | 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: | $2^{2}3^{4/3}7^{2/3}\approx 63.33158505484892$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $36$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(504=2^{3}\cdot 3^{2}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(403,·)$, $\chi_{504}(277,·)$, $\chi_{504}(151,·)$, $\chi_{504}(25,·)$, $\chi_{504}(415,·)$, $\chi_{504}(289,·)$, $\chi_{504}(163,·)$, $\chi_{504}(37,·)$, $\chi_{504}(295,·)$, $\chi_{504}(169,·)$, $\chi_{504}(43,·)$, $\chi_{504}(445,·)$, $\chi_{504}(319,·)$, $\chi_{504}(193,·)$, $\chi_{504}(67,·)$, $\chi_{504}(457,·)$, $\chi_{504}(331,·)$, $\chi_{504}(205,·)$, $\chi_{504}(79,·)$, $\chi_{504}(337,·)$, $\chi_{504}(211,·)$, $\chi_{504}(85,·)$, $\chi_{504}(463,·)$, $\chi_{504}(421,·)$, $\chi_{504}(487,·)$, $\chi_{504}(361,·)$, $\chi_{504}(235,·)$, $\chi_{504}(109,·)$, $\chi_{504}(499,·)$, $\chi_{504}(373,·)$, $\chi_{504}(247,·)$, $\chi_{504}(121,·)$, $\chi_{504}(379,·)$, $\chi_{504}(253,·)$, $\chi_{504}(127,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{1537}a^{24}+\frac{139}{1537}a^{20}-\frac{33}{1537}a^{16}+\frac{340}{1537}a^{12}+\frac{723}{1537}a^{8}+\frac{150}{1537}a^{4}+\frac{49}{1537}$, $\frac{1}{1537}a^{25}+\frac{139}{1537}a^{21}-\frac{33}{1537}a^{17}+\frac{340}{1537}a^{13}+\frac{723}{1537}a^{9}+\frac{150}{1537}a^{5}+\frac{49}{1537}a$, $\frac{1}{1537}a^{26}+\frac{139}{1537}a^{22}-\frac{33}{1537}a^{18}+\frac{340}{1537}a^{14}+\frac{723}{1537}a^{10}+\frac{150}{1537}a^{6}+\frac{49}{1537}a^{2}$, $\frac{1}{1537}a^{27}+\frac{139}{1537}a^{23}-\frac{33}{1537}a^{19}+\frac{340}{1537}a^{15}+\frac{723}{1537}a^{11}+\frac{150}{1537}a^{7}+\frac{49}{1537}a^{3}$, $\frac{1}{12296}a^{28}-\frac{498}{1537}a^{20}-\frac{729}{1537}a^{16}+\frac{523}{1537}a^{12}+\frac{329}{1537}a^{8}+\frac{666}{1537}a^{4}-\frac{663}{12296}$, $\frac{1}{12296}a^{29}-\frac{498}{1537}a^{21}-\frac{729}{1537}a^{17}+\frac{523}{1537}a^{13}+\frac{329}{1537}a^{9}+\frac{666}{1537}a^{5}-\frac{663}{12296}a$, $\frac{1}{12296}a^{30}-\frac{498}{1537}a^{22}-\frac{729}{1537}a^{18}+\frac{523}{1537}a^{14}+\frac{329}{1537}a^{10}+\frac{666}{1537}a^{6}-\frac{663}{12296}a^{2}$, $\frac{1}{12296}a^{31}-\frac{498}{1537}a^{23}-\frac{729}{1537}a^{19}+\frac{523}{1537}a^{15}+\frac{329}{1537}a^{11}+\frac{666}{1537}a^{7}-\frac{663}{12296}a^{3}$, $\frac{1}{29\!\cdots\!84}a^{32}+\frac{70\!\cdots\!73}{29\!\cdots\!84}a^{28}+\frac{80\!\cdots\!05}{37\!\cdots\!48}a^{24}-\frac{22\!\cdots\!47}{37\!\cdots\!48}a^{20}+\frac{83\!\cdots\!41}{18\!\cdots\!24}a^{16}+\frac{53\!\cdots\!65}{46\!\cdots\!31}a^{12}+\frac{17\!\cdots\!47}{37\!\cdots\!48}a^{8}+\frac{53\!\cdots\!69}{29\!\cdots\!84}a^{4}+\frac{73\!\cdots\!53}{29\!\cdots\!84}$, $\frac{1}{29\!\cdots\!84}a^{33}+\frac{70\!\cdots\!73}{29\!\cdots\!84}a^{29}+\frac{80\!\cdots\!05}{37\!\cdots\!48}a^{25}-\frac{22\!\cdots\!47}{37\!\cdots\!48}a^{21}+\frac{83\!\cdots\!41}{18\!\cdots\!24}a^{17}+\frac{53\!\cdots\!65}{46\!\cdots\!31}a^{13}+\frac{17\!\cdots\!47}{37\!\cdots\!48}a^{9}+\frac{53\!\cdots\!69}{29\!\cdots\!84}a^{5}+\frac{73\!\cdots\!53}{29\!\cdots\!84}a$, $\frac{1}{29\!\cdots\!84}a^{34}+\frac{70\!\cdots\!73}{29\!\cdots\!84}a^{30}+\frac{80\!\cdots\!05}{37\!\cdots\!48}a^{26}-\frac{22\!\cdots\!47}{37\!\cdots\!48}a^{22}+\frac{83\!\cdots\!41}{18\!\cdots\!24}a^{18}+\frac{53\!\cdots\!65}{46\!\cdots\!31}a^{14}+\frac{17\!\cdots\!47}{37\!\cdots\!48}a^{10}+\frac{53\!\cdots\!69}{29\!\cdots\!84}a^{6}+\frac{73\!\cdots\!53}{29\!\cdots\!84}a^{2}$, $\frac{1}{29\!\cdots\!84}a^{35}+\frac{70\!\cdots\!73}{29\!\cdots\!84}a^{31}+\frac{80\!\cdots\!05}{37\!\cdots\!48}a^{27}-\frac{22\!\cdots\!47}{37\!\cdots\!48}a^{23}+\frac{83\!\cdots\!41}{18\!\cdots\!24}a^{19}+\frac{53\!\cdots\!65}{46\!\cdots\!31}a^{15}+\frac{17\!\cdots\!47}{37\!\cdots\!48}a^{11}+\frac{53\!\cdots\!69}{29\!\cdots\!84}a^{7}+\frac{73\!\cdots\!53}{29\!\cdots\!84}a^{3}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{2}\times C_{18}\times C_{126}$, which has order $9072$ (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: | $17$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{777169578454749605}{116544808121121688096} a^{33} - \frac{181842965177485526397}{116544808121121688096} a^{29} - \frac{1602190315831148653893}{14568101015140211012} a^{25} - \frac{41071031422755870665389}{14568101015140211012} a^{21} - \frac{166692176757014898869253}{7284050507570105506} a^{17} - \frac{67564173576078929817462}{3642025253785052753} a^{13} - \frac{68551006951128593547867}{14568101015140211012} a^{9} - \frac{46568446236894731217381}{116544808121121688096} a^{5} - \frac{780433761781366821813}{116544808121121688096} a \)
(order $8$)
| 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: |
$\frac{83\!\cdots\!69}{55\!\cdots\!28}a^{32}+\frac{19\!\cdots\!89}{55\!\cdots\!28}a^{28}+\frac{17\!\cdots\!01}{69\!\cdots\!16}a^{24}+\frac{44\!\cdots\!97}{69\!\cdots\!16}a^{20}+\frac{18\!\cdots\!09}{34\!\cdots\!08}a^{16}+\frac{40\!\cdots\!59}{87\!\cdots\!27}a^{12}+\frac{96\!\cdots\!03}{69\!\cdots\!16}a^{8}+\frac{73\!\cdots\!17}{55\!\cdots\!28}a^{4}+\frac{12\!\cdots\!01}{55\!\cdots\!28}$, $\frac{83\!\cdots\!69}{55\!\cdots\!28}a^{32}+\frac{19\!\cdots\!89}{55\!\cdots\!28}a^{28}+\frac{17\!\cdots\!01}{69\!\cdots\!16}a^{24}+\frac{44\!\cdots\!97}{69\!\cdots\!16}a^{20}+\frac{18\!\cdots\!09}{34\!\cdots\!08}a^{16}+\frac{40\!\cdots\!59}{87\!\cdots\!27}a^{12}+\frac{96\!\cdots\!03}{69\!\cdots\!16}a^{8}+\frac{73\!\cdots\!17}{55\!\cdots\!28}a^{4}+\frac{73\!\cdots\!73}{55\!\cdots\!28}$, $\frac{17\!\cdots\!79}{29\!\cdots\!84}a^{34}+\frac{39\!\cdots\!15}{29\!\cdots\!84}a^{30}+\frac{35\!\cdots\!95}{37\!\cdots\!48}a^{26}+\frac{90\!\cdots\!23}{37\!\cdots\!48}a^{22}+\frac{36\!\cdots\!63}{18\!\cdots\!24}a^{18}+\frac{80\!\cdots\!65}{46\!\cdots\!31}a^{14}+\frac{18\!\cdots\!25}{37\!\cdots\!48}a^{10}+\frac{15\!\cdots\!79}{29\!\cdots\!84}a^{6}+\frac{43\!\cdots\!03}{29\!\cdots\!84}a^{2}$, $\frac{79\!\cdots\!39}{29\!\cdots\!84}a^{35}+\frac{18\!\cdots\!95}{29\!\cdots\!84}a^{31}+\frac{16\!\cdots\!47}{37\!\cdots\!48}a^{27}+\frac{42\!\cdots\!23}{37\!\cdots\!48}a^{23}+\frac{17\!\cdots\!39}{18\!\cdots\!24}a^{19}+\frac{35\!\cdots\!30}{46\!\cdots\!31}a^{15}+\frac{73\!\cdots\!69}{37\!\cdots\!48}a^{11}+\frac{19\!\cdots\!95}{10\!\cdots\!96}a^{7}+\frac{12\!\cdots\!99}{29\!\cdots\!84}a^{3}$, $\frac{20\!\cdots\!85}{11\!\cdots\!96}a^{35}-\frac{33442029650001}{16\!\cdots\!44}a^{34}+\frac{77\!\cdots\!05}{11\!\cdots\!96}a^{33}+\frac{46\!\cdots\!29}{11\!\cdots\!96}a^{31}-\frac{78\!\cdots\!37}{16\!\cdots\!44}a^{30}+\frac{18\!\cdots\!97}{11\!\cdots\!96}a^{29}+\frac{41\!\cdots\!25}{14\!\cdots\!12}a^{27}-\frac{68\!\cdots\!25}{204367053127493}a^{26}+\frac{16\!\cdots\!93}{14\!\cdots\!12}a^{25}+\frac{10\!\cdots\!45}{14\!\cdots\!12}a^{23}-\frac{17\!\cdots\!49}{204367053127493}a^{22}+\frac{41\!\cdots\!89}{14\!\cdots\!12}a^{21}+\frac{43\!\cdots\!25}{72\!\cdots\!06}a^{19}-\frac{14\!\cdots\!66}{204367053127493}a^{18}+\frac{16\!\cdots\!53}{72\!\cdots\!06}a^{17}+\frac{17\!\cdots\!00}{36\!\cdots\!53}a^{15}-\frac{11\!\cdots\!11}{204367053127493}a^{14}+\frac{67\!\cdots\!62}{36\!\cdots\!53}a^{13}+\frac{18\!\cdots\!95}{14\!\cdots\!12}a^{11}-\frac{32\!\cdots\!59}{204367053127493}a^{10}+\frac{68\!\cdots\!67}{14\!\cdots\!12}a^{9}+\frac{14\!\cdots\!85}{11\!\cdots\!96}a^{7}-\frac{25\!\cdots\!61}{16\!\cdots\!44}a^{6}+\frac{46\!\cdots\!81}{11\!\cdots\!96}a^{5}+\frac{32\!\cdots\!45}{11\!\cdots\!96}a^{3}-\frac{69\!\cdots\!73}{16\!\cdots\!44}a^{2}+\frac{78\!\cdots\!13}{11\!\cdots\!96}a$, $\frac{33442029650001}{16\!\cdots\!44}a^{34}+\frac{26\!\cdots\!91}{29\!\cdots\!84}a^{33}+\frac{78\!\cdots\!37}{16\!\cdots\!44}a^{30}+\frac{61\!\cdots\!23}{29\!\cdots\!84}a^{29}+\frac{68\!\cdots\!25}{204367053127493}a^{26}+\frac{54\!\cdots\!27}{37\!\cdots\!48}a^{25}+\frac{17\!\cdots\!49}{204367053127493}a^{22}+\frac{13\!\cdots\!83}{37\!\cdots\!48}a^{21}+\frac{14\!\cdots\!66}{204367053127493}a^{18}+\frac{53\!\cdots\!99}{18\!\cdots\!24}a^{17}+\frac{11\!\cdots\!11}{204367053127493}a^{14}+\frac{48\!\cdots\!09}{46\!\cdots\!31}a^{13}+\frac{32\!\cdots\!59}{204367053127493}a^{10}-\frac{13\!\cdots\!07}{37\!\cdots\!48}a^{9}+\frac{25\!\cdots\!61}{16\!\cdots\!44}a^{6}-\frac{37\!\cdots\!05}{29\!\cdots\!84}a^{5}+\frac{69\!\cdots\!73}{16\!\cdots\!44}a^{2}-\frac{24\!\cdots\!01}{29\!\cdots\!84}a+1$, $\frac{20\!\cdots\!85}{11\!\cdots\!96}a^{35}-\frac{30\!\cdots\!11}{14\!\cdots\!92}a^{34}+\frac{77\!\cdots\!05}{11\!\cdots\!96}a^{33}+\frac{46\!\cdots\!29}{11\!\cdots\!96}a^{31}-\frac{72\!\cdots\!35}{14\!\cdots\!92}a^{30}+\frac{18\!\cdots\!97}{11\!\cdots\!96}a^{29}+\frac{41\!\cdots\!25}{14\!\cdots\!12}a^{27}-\frac{63\!\cdots\!27}{18\!\cdots\!24}a^{26}+\frac{16\!\cdots\!93}{14\!\cdots\!12}a^{25}+\frac{10\!\cdots\!45}{14\!\cdots\!12}a^{23}-\frac{16\!\cdots\!71}{18\!\cdots\!24}a^{22}+\frac{41\!\cdots\!89}{14\!\cdots\!12}a^{21}+\frac{43\!\cdots\!25}{72\!\cdots\!06}a^{19}-\frac{66\!\cdots\!71}{92\!\cdots\!62}a^{18}+\frac{16\!\cdots\!53}{72\!\cdots\!06}a^{17}+\frac{17\!\cdots\!00}{36\!\cdots\!53}a^{15}-\frac{26\!\cdots\!36}{46\!\cdots\!31}a^{14}+\frac{67\!\cdots\!62}{36\!\cdots\!53}a^{13}+\frac{18\!\cdots\!95}{14\!\cdots\!12}a^{11}-\frac{26\!\cdots\!81}{18\!\cdots\!24}a^{10}+\frac{68\!\cdots\!67}{14\!\cdots\!12}a^{9}+\frac{14\!\cdots\!85}{11\!\cdots\!96}a^{7}-\frac{17\!\cdots\!03}{14\!\cdots\!92}a^{6}+\frac{46\!\cdots\!81}{11\!\cdots\!96}a^{5}+\frac{32\!\cdots\!45}{11\!\cdots\!96}a^{3}-\frac{28\!\cdots\!31}{14\!\cdots\!92}a^{2}+\frac{78\!\cdots\!13}{11\!\cdots\!96}a$, $\frac{39\!\cdots\!13}{29\!\cdots\!84}a^{35}+\frac{93\!\cdots\!49}{29\!\cdots\!84}a^{31}+\frac{82\!\cdots\!13}{37\!\cdots\!48}a^{27}+\frac{21\!\cdots\!37}{37\!\cdots\!48}a^{23}+\frac{85\!\cdots\!93}{18\!\cdots\!24}a^{19}+\frac{17\!\cdots\!06}{46\!\cdots\!31}a^{15}+\frac{37\!\cdots\!15}{37\!\cdots\!48}a^{11}+\frac{28\!\cdots\!57}{29\!\cdots\!84}a^{7}+\frac{66\!\cdots\!93}{29\!\cdots\!84}a^{3}$, $\frac{49\!\cdots\!83}{74\!\cdots\!96}a^{35}+\frac{11\!\cdots\!53}{74\!\cdots\!96}a^{31}+\frac{10\!\cdots\!93}{92\!\cdots\!62}a^{27}+\frac{26\!\cdots\!11}{92\!\cdots\!62}a^{23}+\frac{10\!\cdots\!54}{46\!\cdots\!31}a^{19}+\frac{88\!\cdots\!11}{46\!\cdots\!31}a^{15}+\frac{47\!\cdots\!51}{92\!\cdots\!62}a^{11}+\frac{37\!\cdots\!07}{74\!\cdots\!96}a^{7}+\frac{94\!\cdots\!33}{74\!\cdots\!96}a^{3}$, $\frac{75\!\cdots\!57}{29\!\cdots\!84}a^{32}+\frac{17\!\cdots\!97}{29\!\cdots\!84}a^{28}+\frac{15\!\cdots\!49}{37\!\cdots\!48}a^{24}+\frac{39\!\cdots\!13}{37\!\cdots\!48}a^{20}+\frac{16\!\cdots\!49}{18\!\cdots\!24}a^{16}+\frac{32\!\cdots\!92}{46\!\cdots\!31}a^{12}+\frac{63\!\cdots\!75}{37\!\cdots\!48}a^{8}+\frac{41\!\cdots\!37}{29\!\cdots\!84}a^{4}+\frac{94\!\cdots\!97}{29\!\cdots\!84}$, $\frac{81\!\cdots\!05}{14\!\cdots\!92}a^{35}+\frac{19\!\cdots\!17}{14\!\cdots\!92}a^{31}+\frac{16\!\cdots\!69}{18\!\cdots\!24}a^{27}+\frac{43\!\cdots\!49}{18\!\cdots\!24}a^{23}+\frac{17\!\cdots\!95}{92\!\cdots\!62}a^{19}+\frac{72\!\cdots\!49}{46\!\cdots\!31}a^{15}+\frac{77\!\cdots\!51}{18\!\cdots\!24}a^{11}+\frac{58\!\cdots\!09}{14\!\cdots\!92}a^{7}+\frac{13\!\cdots\!13}{14\!\cdots\!92}a^{3}$, $\frac{33442029650001}{16\!\cdots\!44}a^{34}-\frac{37\!\cdots\!95}{29\!\cdots\!84}a^{33}+\frac{78\!\cdots\!37}{16\!\cdots\!44}a^{30}-\frac{88\!\cdots\!15}{29\!\cdots\!84}a^{29}+\frac{68\!\cdots\!25}{204367053127493}a^{26}-\frac{77\!\cdots\!79}{37\!\cdots\!48}a^{25}+\frac{17\!\cdots\!49}{204367053127493}a^{22}-\frac{19\!\cdots\!95}{37\!\cdots\!48}a^{21}+\frac{14\!\cdots\!66}{204367053127493}a^{18}-\frac{80\!\cdots\!03}{18\!\cdots\!24}a^{17}+\frac{11\!\cdots\!11}{204367053127493}a^{14}-\frac{15\!\cdots\!35}{46\!\cdots\!31}a^{13}+\frac{32\!\cdots\!59}{204367053127493}a^{10}-\frac{29\!\cdots\!85}{37\!\cdots\!48}a^{9}+\frac{25\!\cdots\!61}{16\!\cdots\!44}a^{6}-\frac{64\!\cdots\!39}{10\!\cdots\!96}a^{5}+\frac{69\!\cdots\!73}{16\!\cdots\!44}a^{2}-\frac{30\!\cdots\!07}{29\!\cdots\!84}a+1$, $\frac{20\!\cdots\!85}{11\!\cdots\!96}a^{35}-\frac{77\!\cdots\!05}{11\!\cdots\!96}a^{33}+\frac{64\!\cdots\!89}{10\!\cdots\!96}a^{32}+\frac{46\!\cdots\!29}{11\!\cdots\!96}a^{31}-\frac{18\!\cdots\!97}{11\!\cdots\!96}a^{29}+\frac{15\!\cdots\!05}{10\!\cdots\!96}a^{28}+\frac{41\!\cdots\!25}{14\!\cdots\!12}a^{27}-\frac{16\!\cdots\!93}{14\!\cdots\!12}a^{25}+\frac{13\!\cdots\!01}{12\!\cdots\!12}a^{24}+\frac{10\!\cdots\!45}{14\!\cdots\!12}a^{23}-\frac{41\!\cdots\!89}{14\!\cdots\!12}a^{21}+\frac{33\!\cdots\!93}{12\!\cdots\!12}a^{20}+\frac{43\!\cdots\!25}{72\!\cdots\!06}a^{19}-\frac{16\!\cdots\!53}{72\!\cdots\!06}a^{17}+\frac{13\!\cdots\!33}{63\!\cdots\!56}a^{16}+\frac{17\!\cdots\!00}{36\!\cdots\!53}a^{15}-\frac{67\!\cdots\!62}{36\!\cdots\!53}a^{13}+\frac{20\!\cdots\!96}{15\!\cdots\!39}a^{12}+\frac{18\!\cdots\!95}{14\!\cdots\!12}a^{11}-\frac{68\!\cdots\!67}{14\!\cdots\!12}a^{9}+\frac{23\!\cdots\!87}{12\!\cdots\!12}a^{8}+\frac{14\!\cdots\!85}{11\!\cdots\!96}a^{7}-\frac{46\!\cdots\!81}{11\!\cdots\!96}a^{5}+\frac{51\!\cdots\!09}{10\!\cdots\!96}a^{4}+\frac{32\!\cdots\!45}{11\!\cdots\!96}a^{3}-\frac{78\!\cdots\!13}{11\!\cdots\!96}a-\frac{35\!\cdots\!27}{10\!\cdots\!96}$, $\frac{32\!\cdots\!83}{74\!\cdots\!96}a^{35}-\frac{10\!\cdots\!97}{92\!\cdots\!62}a^{33}-\frac{64\!\cdots\!05}{63\!\cdots\!56}a^{32}+\frac{76\!\cdots\!33}{74\!\cdots\!96}a^{31}-\frac{10\!\cdots\!49}{37\!\cdots\!48}a^{29}-\frac{87\!\cdots\!93}{37\!\cdots\!48}a^{28}+\frac{67\!\cdots\!71}{92\!\cdots\!62}a^{27}-\frac{90\!\cdots\!44}{46\!\cdots\!31}a^{25}-\frac{77\!\cdots\!16}{46\!\cdots\!31}a^{24}+\frac{17\!\cdots\!65}{92\!\cdots\!62}a^{23}-\frac{23\!\cdots\!39}{46\!\cdots\!31}a^{21}-\frac{19\!\cdots\!67}{46\!\cdots\!31}a^{20}+\frac{70\!\cdots\!25}{46\!\cdots\!31}a^{19}-\frac{18\!\cdots\!51}{46\!\cdots\!31}a^{17}-\frac{16\!\cdots\!12}{46\!\cdots\!31}a^{16}+\frac{58\!\cdots\!77}{46\!\cdots\!31}a^{15}-\frac{15\!\cdots\!26}{46\!\cdots\!31}a^{13}-\frac{15\!\cdots\!62}{46\!\cdots\!31}a^{12}+\frac{30\!\cdots\!45}{92\!\cdots\!62}a^{11}-\frac{38\!\cdots\!74}{46\!\cdots\!31}a^{9}-\frac{51\!\cdots\!09}{46\!\cdots\!31}a^{8}+\frac{23\!\cdots\!67}{74\!\cdots\!96}a^{7}-\frac{70\!\cdots\!23}{92\!\cdots\!62}a^{5}-\frac{24\!\cdots\!33}{18\!\cdots\!24}a^{4}+\frac{57\!\cdots\!33}{74\!\cdots\!96}a^{3}-\frac{68\!\cdots\!01}{37\!\cdots\!48}a-\frac{14\!\cdots\!25}{37\!\cdots\!48}$, $\frac{85\!\cdots\!43}{14\!\cdots\!92}a^{35}-\frac{46\!\cdots\!83}{74\!\cdots\!96}a^{34}+\frac{91\!\cdots\!43}{92\!\cdots\!62}a^{32}+\frac{19\!\cdots\!59}{14\!\cdots\!92}a^{31}-\frac{10\!\cdots\!61}{74\!\cdots\!96}a^{30}+\frac{85\!\cdots\!71}{37\!\cdots\!48}a^{28}+\frac{17\!\cdots\!87}{18\!\cdots\!24}a^{27}-\frac{96\!\cdots\!91}{92\!\cdots\!62}a^{26}+\frac{75\!\cdots\!03}{46\!\cdots\!31}a^{24}+\frac{44\!\cdots\!27}{18\!\cdots\!24}a^{23}-\frac{24\!\cdots\!89}{92\!\cdots\!62}a^{22}+\frac{19\!\cdots\!20}{46\!\cdots\!31}a^{20}+\frac{18\!\cdots\!53}{92\!\cdots\!62}a^{19}-\frac{10\!\cdots\!36}{46\!\cdots\!31}a^{18}+\frac{15\!\cdots\!47}{46\!\cdots\!31}a^{16}+\frac{75\!\cdots\!78}{46\!\cdots\!31}a^{15}-\frac{83\!\cdots\!59}{46\!\cdots\!31}a^{14}+\frac{11\!\cdots\!14}{46\!\cdots\!31}a^{12}+\frac{80\!\cdots\!05}{18\!\cdots\!24}a^{11}-\frac{44\!\cdots\!23}{92\!\cdots\!62}a^{10}+\frac{28\!\cdots\!38}{46\!\cdots\!31}a^{8}+\frac{21\!\cdots\!59}{51\!\cdots\!48}a^{7}-\frac{35\!\cdots\!07}{74\!\cdots\!96}a^{6}+\frac{73\!\cdots\!23}{92\!\cdots\!62}a^{4}+\frac{14\!\cdots\!79}{14\!\cdots\!92}a^{3}-\frac{87\!\cdots\!25}{74\!\cdots\!96}a^{2}+\frac{10\!\cdots\!03}{37\!\cdots\!48}$, $\frac{49\!\cdots\!83}{74\!\cdots\!96}a^{35}-\frac{38\!\cdots\!37}{29\!\cdots\!84}a^{34}+\frac{23\!\cdots\!23}{37\!\cdots\!48}a^{33}+\frac{11\!\cdots\!53}{74\!\cdots\!96}a^{31}-\frac{89\!\cdots\!05}{29\!\cdots\!84}a^{30}+\frac{26\!\cdots\!25}{18\!\cdots\!24}a^{29}+\frac{10\!\cdots\!93}{92\!\cdots\!62}a^{27}-\frac{78\!\cdots\!05}{37\!\cdots\!48}a^{26}+\frac{47\!\cdots\!46}{46\!\cdots\!31}a^{25}+\frac{26\!\cdots\!11}{92\!\cdots\!62}a^{23}-\frac{20\!\cdots\!21}{37\!\cdots\!48}a^{22}+\frac{12\!\cdots\!91}{46\!\cdots\!31}a^{21}+\frac{10\!\cdots\!54}{46\!\cdots\!31}a^{19}-\frac{82\!\cdots\!41}{18\!\cdots\!24}a^{18}+\frac{10\!\cdots\!02}{46\!\cdots\!31}a^{17}+\frac{88\!\cdots\!11}{46\!\cdots\!31}a^{15}-\frac{17\!\cdots\!88}{46\!\cdots\!31}a^{14}+\frac{92\!\cdots\!75}{46\!\cdots\!31}a^{13}+\frac{47\!\cdots\!51}{92\!\cdots\!62}a^{11}-\frac{37\!\cdots\!71}{37\!\cdots\!48}a^{10}+\frac{29\!\cdots\!10}{46\!\cdots\!31}a^{9}+\frac{37\!\cdots\!07}{74\!\cdots\!96}a^{7}-\frac{29\!\cdots\!01}{29\!\cdots\!84}a^{6}+\frac{28\!\cdots\!83}{37\!\cdots\!48}a^{5}+\frac{94\!\cdots\!33}{74\!\cdots\!96}a^{3}-\frac{81\!\cdots\!93}{29\!\cdots\!84}a^{2}+\frac{55\!\cdots\!73}{18\!\cdots\!24}a$, $\frac{26\!\cdots\!57}{29\!\cdots\!84}a^{35}-\frac{26\!\cdots\!45}{29\!\cdots\!84}a^{34}-\frac{57\!\cdots\!91}{29\!\cdots\!84}a^{32}+\frac{62\!\cdots\!77}{29\!\cdots\!84}a^{31}-\frac{62\!\cdots\!41}{29\!\cdots\!84}a^{30}-\frac{13\!\cdots\!03}{29\!\cdots\!84}a^{28}+\frac{55\!\cdots\!81}{37\!\cdots\!48}a^{27}-\frac{55\!\cdots\!41}{37\!\cdots\!48}a^{26}-\frac{11\!\cdots\!23}{37\!\cdots\!48}a^{24}+\frac{14\!\cdots\!81}{37\!\cdots\!48}a^{23}-\frac{14\!\cdots\!81}{37\!\cdots\!48}a^{22}-\frac{30\!\cdots\!55}{37\!\cdots\!48}a^{20}+\frac{57\!\cdots\!57}{18\!\cdots\!24}a^{19}-\frac{57\!\cdots\!13}{18\!\cdots\!24}a^{18}-\frac{11\!\cdots\!71}{18\!\cdots\!24}a^{16}+\frac{11\!\cdots\!68}{46\!\cdots\!31}a^{15}-\frac{12\!\cdots\!82}{46\!\cdots\!31}a^{14}-\frac{15\!\cdots\!74}{46\!\cdots\!31}a^{12}+\frac{25\!\cdots\!31}{37\!\cdots\!48}a^{11}-\frac{26\!\cdots\!79}{37\!\cdots\!48}a^{10}+\frac{10\!\cdots\!91}{37\!\cdots\!48}a^{8}+\frac{19\!\cdots\!49}{29\!\cdots\!84}a^{7}-\frac{21\!\cdots\!77}{29\!\cdots\!84}a^{6}+\frac{34\!\cdots\!69}{29\!\cdots\!84}a^{4}+\frac{46\!\cdots\!61}{29\!\cdots\!84}a^{3}-\frac{53\!\cdots\!69}{29\!\cdots\!84}a^{2}+\frac{12\!\cdots\!05}{29\!\cdots\!84}$
| 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: | \( 160258501280890.78 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 160258501280890.78 \cdot 9072}{8\cdot\sqrt{72166039996680932841568261508505246616460162082637588781812678656}}\cr\approx \mathstrut & 0.157581534269550 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 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^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 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^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 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^36 + 234*x^32 + 16497*x^28 + 423088*x^24 + 3439800*x^20 + 2847312*x^16 + 761144*x^12 + 73593*x^8 + 1794*x^4 + 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
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 36 |
| The 36 conjugacy class representatives for $C_6^2$ |
| Character table for $C_6^2$ is not computed |
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 | ${\href{/padicField/5.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | ${\href{/padicField/13.6.0.1}{6} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{12}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.6.0.1}{6} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.6.0.1}{6} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{12}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.6.0.1}{6} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.12.24.79 | $x^{12} - 8 x^{11} + 14 x^{10} + 76 x^{9} + 138 x^{8} + 432 x^{7} + 688 x^{6} + 992 x^{5} + 1748 x^{4} + 1728 x^{3} + 1848 x^{2} + 1648 x + 968$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ |
| 2.12.24.79 | $x^{12} - 8 x^{11} + 14 x^{10} + 76 x^{9} + 138 x^{8} + 432 x^{7} + 688 x^{6} + 992 x^{5} + 1748 x^{4} + 1728 x^{3} + 1848 x^{2} + 1648 x + 968$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ | |
| 2.12.24.79 | $x^{12} - 8 x^{11} + 14 x^{10} + 76 x^{9} + 138 x^{8} + 432 x^{7} + 688 x^{6} + 992 x^{5} + 1748 x^{4} + 1728 x^{3} + 1848 x^{2} + 1648 x + 968$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ | |
|
\(3\)
| Deg $18$ | $3$ | $6$ | $24$ | |||
| Deg $18$ | $3$ | $6$ | $24$ | ||||
|
\(7\)
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
| 7.18.12.1 | $x^{18} + 3 x^{16} + 57 x^{15} + 15 x^{14} + 90 x^{13} + 424 x^{12} - 921 x^{11} - 3090 x^{10} - 6496 x^{9} - 10560 x^{8} + 6912 x^{7} + 28033 x^{6} + 33237 x^{5} + 188463 x^{4} - 139476 x^{3} + 351552 x^{2} - 514905 x + 582014$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |