Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*x^2 + 1)
gp: K = bnfinit(y^36 - 9*y^34 + 67*y^32 - 478*y^30 + 3373*y^28 - 23732*y^26 + 166844*y^24 - 325502*y^22 + 617434*y^20 - 1166722*y^18 + 2181924*y^16 - 3920642*y^14 + 5905564*y^12 - 443012*y^10 + 33233*y^8 - 2493*y^6 + 187*y^4 - 14*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*x^2 + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*x^2 + 1)
\( x^{36} - 9 x^{34} + 67 x^{32} - 478 x^{30} + 3373 x^{28} - 23732 x^{26} + 166844 x^{24} - 325502 x^{22} + \cdots + 1 \)
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: |
\(840739592110096304569466677876458557457237523263143378686050304\)
\(\medspace = 2^{36}\cdot 7^{30}\cdot 13^{24}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(55.96\) | 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: | $2\cdot 7^{5/6}13^{2/3}\approx 55.96380876440059$ | ||
| Ramified primes: |
\(2\), \(7\), \(13\)
| 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: | \(364=2^{2}\cdot 7\cdot 13\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{364}(1,·)$, $\chi_{364}(3,·)$, $\chi_{364}(261,·)$, $\chi_{364}(263,·)$, $\chi_{364}(9,·)$, $\chi_{364}(139,·)$, $\chi_{364}(269,·)$, $\chi_{364}(131,·)$, $\chi_{364}(27,·)$, $\chi_{364}(29,·)$, $\chi_{364}(159,·)$, $\chi_{364}(289,·)$, $\chi_{364}(165,·)$, $\chi_{364}(295,·)$, $\chi_{364}(157,·)$, $\chi_{364}(243,·)$, $\chi_{364}(53,·)$, $\chi_{364}(183,·)$, $\chi_{364}(185,·)$, $\chi_{364}(87,·)$, $\chi_{364}(61,·)$, $\chi_{364}(191,·)$, $\chi_{364}(321,·)$, $\chi_{364}(55,·)$, $\chi_{364}(79,·)$, $\chi_{364}(209,·)$, $\chi_{364}(211,·)$, $\chi_{364}(341,·)$, $\chi_{364}(313,·)$, $\chi_{364}(347,·)$, $\chi_{364}(107,·)$, $\chi_{364}(81,·)$, $\chi_{364}(235,·)$, $\chi_{364}(237,·)$, $\chi_{364}(113,·)$, $\chi_{364}(339,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{131072}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{4}a^{14}+\frac{1}{4}$, $\frac{1}{4}a^{15}+\frac{1}{4}a$, $\frac{1}{4}a^{16}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{17}+\frac{1}{4}a^{3}$, $\frac{1}{4}a^{18}+\frac{1}{4}a^{4}$, $\frac{1}{4}a^{19}+\frac{1}{4}a^{5}$, $\frac{1}{4}a^{20}+\frac{1}{4}a^{6}$, $\frac{1}{4}a^{21}+\frac{1}{4}a^{7}$, $\frac{1}{4}a^{22}+\frac{1}{4}a^{8}$, $\frac{1}{4}a^{23}+\frac{1}{4}a^{9}$, $\frac{1}{20}a^{24}+\frac{1}{20}a^{20}+\frac{1}{20}a^{16}-\frac{1}{5}a^{12}+\frac{1}{4}a^{10}-\frac{1}{5}a^{8}+\frac{1}{4}a^{6}-\frac{1}{5}a^{4}+\frac{1}{4}a^{2}-\frac{1}{5}$, $\frac{1}{20}a^{25}+\frac{1}{20}a^{21}+\frac{1}{20}a^{17}-\frac{1}{5}a^{13}+\frac{1}{4}a^{11}-\frac{1}{5}a^{9}+\frac{1}{4}a^{7}-\frac{1}{5}a^{5}+\frac{1}{4}a^{3}-\frac{1}{5}a$, $\frac{1}{304623237260}a^{26}-\frac{4605546001}{304623237260}a^{24}-\frac{37413186779}{304623237260}a^{22}+\frac{20417278449}{304623237260}a^{20}-\frac{36144391709}{304623237260}a^{18}+\frac{715004559}{304623237260}a^{16}+\frac{5558577139}{76155809315}a^{14}-\frac{86986800951}{304623237260}a^{12}+\frac{62203110071}{304623237260}a^{10}-\frac{80658946771}{304623237260}a^{8}-\frac{134082028419}{304623237260}a^{6}+\frac{65241930359}{304623237260}a^{4}+\frac{75981213051}{304623237260}a^{2}-\frac{23082836599}{76155809315}$, $\frac{1}{304623237260}a^{27}-\frac{4605546001}{304623237260}a^{25}-\frac{37413186779}{304623237260}a^{23}+\frac{20417278449}{304623237260}a^{21}-\frac{36144391709}{304623237260}a^{19}+\frac{715004559}{304623237260}a^{17}+\frac{5558577139}{76155809315}a^{15}-\frac{86986800951}{304623237260}a^{13}+\frac{62203110071}{304623237260}a^{11}-\frac{80658946771}{304623237260}a^{9}-\frac{134082028419}{304623237260}a^{7}+\frac{65241930359}{304623237260}a^{5}+\frac{75981213051}{304623237260}a^{3}-\frac{23082836599}{76155809315}a$, $\frac{1}{1218492949040}a^{28}+\frac{8698764817}{121849294904}a^{14}-\frac{6626441751}{1218492949040}$, $\frac{1}{1218492949040}a^{29}+\frac{8698764817}{121849294904}a^{15}-\frac{6626441751}{1218492949040}a$, $\frac{1}{1218492949040}a^{30}+\frac{8698764817}{121849294904}a^{16}-\frac{6626441751}{1218492949040}a^{2}$, $\frac{1}{1218492949040}a^{31}+\frac{8698764817}{121849294904}a^{17}-\frac{6626441751}{1218492949040}a^{3}$, $\frac{1}{1218492949040}a^{32}+\frac{8698764817}{121849294904}a^{18}-\frac{6626441751}{1218492949040}a^{4}$, $\frac{1}{1218492949040}a^{33}+\frac{8698764817}{121849294904}a^{19}-\frac{6626441751}{1218492949040}a^{5}$, $\frac{1}{1218492949040}a^{34}+\frac{8698764817}{121849294904}a^{20}-\frac{6626441751}{1218492949040}a^{6}$, $\frac{1}{1218492949040}a^{35}+\frac{8698764817}{121849294904}a^{21}-\frac{6626441751}{1218492949040}a^{7}$
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
$C_{3}\times C_{6}\times C_{78}$, which has order $1404$ (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: |
\( \frac{11911}{304623237260} a^{29} + \frac{2018324459}{60924647452} a^{15} + \frac{278051973476}{76155809315} a \)
(order $28$)
| 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: |
$\frac{231471117}{60924647452}a^{34}-\frac{1723173871}{60924647452}a^{32}+\frac{6146844107}{30462323726}a^{30}-\frac{86750230849}{60924647452}a^{28}+\frac{152590904129}{15231161863}a^{26}-\frac{1072765751243}{15231161863}a^{24}+\frac{30161226649901}{60924647452}a^{22}-\frac{7939896536321}{30462323726}a^{20}+\frac{15003469142693}{30462323726}a^{18}-\frac{14029232930253}{15231161863}a^{16}+\frac{50417521283173}{30462323726}a^{14}-\frac{37971319322083}{15231161863}a^{12}+\frac{2848457846789}{15231161863}a^{10}+\frac{19\!\cdots\!85}{60924647452}a^{8}+\frac{64117499409}{60924647452}a^{6}-\frac{4809455431}{60924647452}a^{4}+\frac{180033091}{30462323726}a^{2}+\frac{60898928439}{60924647452}$, $\frac{231471117}{60924647452}a^{34}-\frac{6892117463}{243698589808}a^{32}+\frac{6146844107}{30462323726}a^{30}-\frac{86750230849}{60924647452}a^{28}+\frac{152590904129}{15231161863}a^{26}-\frac{1072765751243}{15231161863}a^{24}+\frac{30161226649901}{60924647452}a^{22}-\frac{7939896536321}{30462323726}a^{20}+\frac{60258731794647}{121849294904}a^{18}-\frac{14029232930253}{15231161863}a^{16}+\frac{50417521283173}{30462323726}a^{14}-\frac{37971319322083}{15231161863}a^{12}+\frac{2848457846789}{15231161863}a^{10}+\frac{19\!\cdots\!85}{60924647452}a^{8}+\frac{64117499409}{60924647452}a^{6}+\frac{43286435174213}{243698589808}a^{4}+\frac{180033091}{30462323726}a^{2}-\frac{25719013}{60924647452}$, $\frac{175247656927}{609246474520}a^{35}-\frac{625884489025}{243698589808}a^{33}+\frac{23257867612169}{1218492949040}a^{31}-\frac{33171877918325}{243698589808}a^{29}+\frac{146281722774923}{152311618630}a^{27}-\frac{10\!\cdots\!03}{152311618630}a^{25}+\frac{14\!\cdots\!81}{304623237260}a^{23}-\frac{13\!\cdots\!53}{152311618630}a^{21}+\frac{10\!\cdots\!07}{609246474520}a^{19}-\frac{19\!\cdots\!57}{609246474520}a^{17}+\frac{36\!\cdots\!27}{609246474520}a^{15}-\frac{16\!\cdots\!63}{152311618630}a^{13}+\frac{24\!\cdots\!83}{152311618630}a^{11}-\frac{18\!\cdots\!11}{304623237260}a^{9}+\frac{55703719523225}{121849294904}a^{7}-\frac{41784048487309}{1218492949040}a^{5}+\frac{625884489025}{243698589808}a^{3}-\frac{225318416049}{1218492949040}a-1$, $\frac{333733709829}{1218492949040}a^{35}-\frac{3003603388461}{1218492949040}a^{33}+\frac{22360158558543}{1218492949040}a^{31}-\frac{79762356649131}{609246474520}a^{29}+\frac{281420950817799}{304623237260}a^{27}-\frac{19\!\cdots\!57}{304623237260}a^{25}+\frac{13\!\cdots\!19}{304623237260}a^{23}-\frac{54\!\cdots\!79}{609246474520}a^{21}+\frac{10\!\cdots\!93}{609246474520}a^{19}-\frac{19\!\cdots\!69}{609246474520}a^{17}+\frac{18\!\cdots\!49}{304623237260}a^{15}-\frac{32\!\cdots\!57}{304623237260}a^{13}+\frac{49\!\cdots\!39}{304623237260}a^{11}-\frac{36\!\cdots\!37}{304623237260}a^{9}+\frac{11\!\cdots\!57}{1218492949040}a^{7}-\frac{831998138603697}{1218492949040}a^{5}+\frac{62408203738023}{1218492949040}a^{3}-\frac{2336135968803}{609246474520}a-1$, $\frac{16902486531}{1218492949040}a^{35}-\frac{125829621953}{1218492949040}a^{33}+\frac{448854920101}{609246474520}a^{31}-\frac{6334676341007}{1218492949040}a^{29}+\frac{11142494732047}{304623237260}a^{27}-\frac{78335512854949}{304623237260}a^{25}+\frac{275304179275681}{152311618630}a^{23}-\frac{579787214932303}{609246474520}a^{21}+\frac{10\!\cdots\!99}{609246474520}a^{19}-\frac{10\!\cdots\!79}{304623237260}a^{17}+\frac{36\!\cdots\!39}{609246474520}a^{15}-\frac{27\!\cdots\!69}{304623237260}a^{13}+\frac{208000121196427}{304623237260}a^{11}+\frac{17\!\cdots\!63}{152311618630}a^{9}+\frac{4681988769087}{1218492949040}a^{7}-\frac{351196109033}{1218492949040}a^{5}+\frac{13146378413}{609246474520}a^{3}-\frac{1878054059}{1218492949040}a+1$, $\frac{62942654419}{609246474520}a^{35}-\frac{566483889771}{609246474520}a^{33}+\frac{4217157846073}{609246474520}a^{31}-\frac{15043294406141}{304623237260}a^{29}+\frac{106152786656743}{304623237260}a^{27}-\frac{373438768667927}{152311618630}a^{25}+\frac{26\!\cdots\!09}{152311618630}a^{23}-\frac{10\!\cdots\!69}{304623237260}a^{21}+\frac{19\!\cdots\!23}{304623237260}a^{19}-\frac{36\!\cdots\!59}{304623237260}a^{17}+\frac{34\!\cdots\!39}{152311618630}a^{15}-\frac{12\!\cdots\!89}{304623237260}a^{13}+\frac{92\!\cdots\!29}{152311618630}a^{11}-\frac{69\!\cdots\!07}{152311618630}a^{9}+\frac{20\!\cdots\!27}{609246474520}a^{7}-\frac{156916037466567}{609246474520}a^{5}+\frac{11770276376353}{609246474520}a^{3}-\frac{440598580933}{304623237260}a$, $\frac{5819546457}{304623237260}a^{35}-\frac{43323290291}{304623237260}a^{33}+\frac{154541289247}{152311618630}a^{31}-\frac{2181036688829}{304623237260}a^{29}+\frac{3836374347709}{76155809315}a^{27}-\frac{26971011363103}{76155809315}a^{25}+\frac{379150241117673}{152311618630}a^{23}-\frac{199621435951741}{152311618630}a^{21}+\frac{377210715633553}{152311618630}a^{19}-\frac{352716891212313}{76155809315}a^{17}+\frac{12\!\cdots\!33}{152311618630}a^{15}-\frac{954658445910743}{76155809315}a^{13}+\frac{71614692083569}{76155809315}a^{11}+\frac{24\!\cdots\!39}{152311618630}a^{9}+\frac{1612014368589}{304623237260}a^{7}-\frac{120917243051}{304623237260}a^{5}+\frac{4526313911}{152311618630}a^{3}-\frac{646616273}{304623237260}a$, $\frac{430286969}{304623237260}a^{34}-\frac{2599855043}{243698589808}a^{32}+\frac{92746548583}{1218492949040}a^{30}-\frac{327232865853}{609246474520}a^{28}+\frac{1151184576209}{304623237260}a^{26}-\frac{8093217572603}{304623237260}a^{24}+\frac{56885990434479}{304623237260}a^{22}-\frac{35272757676703}{304623237260}a^{20}+\frac{113523187148083}{609246474520}a^{18}-\frac{211748363085191}{609246474520}a^{16}+\frac{190195197690259}{304623237260}a^{14}-\frac{286465286980243}{304623237260}a^{12}+\frac{21489490202069}{304623237260}a^{10}+\frac{36\!\cdots\!27}{304623237260}a^{8}-\frac{23485629199376}{15231161863}a^{6}+\frac{59268790295173}{1218492949040}a^{4}-\frac{10573058688117}{1218492949040}a^{2}+\frac{229712493947}{609246474520}$, $\frac{4639987629}{1218492949040}a^{34}-\frac{17232686197}{609246474520}a^{32}+\frac{61468452981}{304623237260}a^{30}-\frac{86750230849}{60924647452}a^{28}+\frac{152590904129}{15231161863}a^{26}-\frac{1072765751243}{15231161863}a^{24}+\frac{30161226649901}{60924647452}a^{22}-\frac{30864474757021}{121849294904}a^{20}+\frac{29926665427067}{60924647452}a^{18}-\frac{56114913396553}{60924647452}a^{16}+\frac{50417521283173}{30462323726}a^{14}-\frac{37971319322083}{15231161863}a^{12}+\frac{2848457846789}{15231161863}a^{10}+\frac{19\!\cdots\!85}{60924647452}a^{8}+\frac{791847636213601}{1218492949040}a^{6}-\frac{70538190197493}{609246474520}a^{4}+\frac{557004112407}{152311618630}a^{2}+\frac{60898928439}{60924647452}$, $\frac{4682570607}{1218492949040}a^{34}-\frac{17228455317}{609246474520}a^{32}+\frac{122937043051}{609246474520}a^{30}-\frac{86750230849}{60924647452}a^{28}+\frac{152590904129}{15231161863}a^{26}-\frac{1072765751243}{15231161863}a^{24}+\frac{30161226649901}{60924647452}a^{22}-\frac{27256764039343}{121849294904}a^{20}+\frac{7571278411816}{15231161863}a^{18}-\frac{56103299138699}{60924647452}a^{16}+\frac{50417521283173}{30462323726}a^{14}-\frac{37971319322083}{15231161863}a^{12}+\frac{2848457846789}{15231161863}a^{10}+\frac{19\!\cdots\!85}{60924647452}a^{8}+\frac{39\!\cdots\!03}{1218492949040}a^{6}+\frac{246132807450237}{609246474520}a^{4}+\frac{11188622255279}{609246474520}a^{2}+\frac{60898928439}{60924647452}$, $\frac{242224901083}{1218492949040}a^{34}-\frac{2180940006947}{1218492949040}a^{32}+\frac{16235922501549}{1218492949040}a^{30}-\frac{7239525070553}{76155809315}a^{28}+\frac{204342410649959}{304623237260}a^{26}-\frac{718863636171681}{152311618630}a^{24}+\frac{50\!\cdots\!27}{152311618630}a^{22}-\frac{39\!\cdots\!89}{609246474520}a^{20}+\frac{74\!\cdots\!03}{609246474520}a^{18}-\frac{14\!\cdots\!89}{609246474520}a^{16}+\frac{13\!\cdots\!29}{304623237260}a^{14}-\frac{23\!\cdots\!97}{304623237260}a^{12}+\frac{17\!\cdots\!87}{152311618630}a^{10}-\frac{13\!\cdots\!21}{152311618630}a^{8}+\frac{401070341743843}{1218492949040}a^{6}-\frac{246613151125151}{1218492949040}a^{4}+\frac{57109454302297}{1218492949040}a^{2}-\frac{69036755503}{60924647452}$, $\frac{5757751697}{304623237260}a^{35}-\frac{220033730943}{1218492949040}a^{34}-\frac{34660523611}{243698589808}a^{33}+\frac{1961999185169}{1218492949040}a^{32}+\frac{618164602789}{609246474520}a^{31}-\frac{14577678799387}{1218492949040}a^{30}-\frac{1744829318881}{243698589808}a^{29}+\frac{12993900926963}{152311618630}a^{28}+\frac{3836374347709}{76155809315}a^{27}-\frac{183358840810523}{304623237260}a^{26}-\frac{26971011363103}{76155809315}a^{25}+\frac{322511190227423}{76155809315}a^{24}+\frac{379150241117673}{152311618630}a^{23}-\frac{45\!\cdots\!29}{152311618630}a^{22}-\frac{112899141608638}{76155809315}a^{21}+\frac{34\!\cdots\!49}{609246474520}a^{20}+\frac{15\!\cdots\!57}{609246474520}a^{19}-\frac{64\!\cdots\!31}{609246474520}a^{18}-\frac{352775582112708}{76155809315}a^{17}+\frac{12\!\cdots\!59}{609246474520}a^{16}+\frac{50\!\cdots\!97}{609246474520}a^{15}-\frac{11\!\cdots\!13}{304623237260}a^{14}-\frac{954658445910743}{76155809315}a^{13}+\frac{20\!\cdots\!77}{304623237260}a^{12}+\frac{71614692083569}{76155809315}a^{11}-\frac{76\!\cdots\!32}{76155809315}a^{10}+\frac{24\!\cdots\!39}{152311618630}a^{9}-\frac{12\!\cdots\!39}{152311618630}a^{8}-\frac{46\!\cdots\!21}{304623237260}a^{7}-\frac{347404850944839}{1218492949040}a^{6}-\frac{709373837960983}{1218492949040}a^{5}+\frac{324262748878209}{1218492949040}a^{4}-\frac{40819453362677}{609246474520}a^{3}-\frac{4864350948523}{243698589808}a^{2}+\frac{10573188653847}{1218492949040}a-\frac{269492817319}{304623237260}$, $\frac{822673160227}{1218492949040}a^{35}-\frac{160730322819}{609246474520}a^{34}-\frac{93792583814}{15231161863}a^{33}+\frac{582445493043}{243698589808}a^{32}+\frac{56001247950027}{1218492949040}a^{31}-\frac{2713540336397}{152311618630}a^{30}-\frac{99946603072137}{304623237260}a^{29}+\frac{154925041621879}{1218492949040}a^{28}+\frac{705390923522671}{304623237260}a^{27}-\frac{68332568542083}{76155809315}a^{26}-\frac{49\!\cdots\!93}{304623237260}a^{25}+\frac{480790475646342}{76155809315}a^{24}+\frac{17\!\cdots\!23}{152311618630}a^{23}-\frac{67\!\cdots\!81}{152311618630}a^{22}-\frac{14\!\cdots\!91}{609246474520}a^{21}+\frac{67\!\cdots\!12}{76155809315}a^{20}+\frac{67\!\cdots\!33}{152311618630}a^{19}-\frac{10\!\cdots\!39}{609246474520}a^{18}-\frac{50\!\cdots\!11}{609246474520}a^{17}+\frac{48\!\cdots\!19}{152311618630}a^{16}+\frac{11\!\cdots\!89}{76155809315}a^{15}-\frac{36\!\cdots\!29}{609246474520}a^{14}-\frac{85\!\cdots\!93}{304623237260}a^{13}+\frac{81\!\cdots\!12}{76155809315}a^{12}+\frac{13\!\cdots\!91}{304623237260}a^{11}-\frac{12\!\cdots\!23}{76155809315}a^{10}-\frac{11\!\cdots\!19}{152311618630}a^{9}+\frac{31\!\cdots\!69}{152311618630}a^{8}+\frac{14\!\cdots\!67}{1218492949040}a^{7}-\frac{11\!\cdots\!25}{121849294904}a^{6}-\frac{176732155853589}{152311618630}a^{5}+\frac{100482919427587}{1218492949040}a^{4}+\frac{365639245719967}{1218492949040}a^{3}-\frac{4914513246777}{76155809315}a^{2}-\frac{3094943098871}{304623237260}a+\frac{7901246890383}{1218492949040}$, $\frac{80151838399}{304623237260}a^{35}-\frac{61518550457}{1218492949040}a^{34}-\frac{2853573849333}{1218492949040}a^{33}+\frac{546810000643}{1218492949040}a^{32}+\frac{21193661693653}{1218492949040}a^{31}-\frac{4060030299389}{1218492949040}a^{30}-\frac{30222705750691}{243698589808}a^{29}+\frac{28946451217977}{1218492949040}a^{28}+\frac{266541017143523}{304623237260}a^{27}-\frac{51056111761517}{304623237260}a^{26}-\frac{18\!\cdots\!07}{304623237260}a^{25}+\frac{89801858458299}{76155809315}a^{24}+\frac{13\!\cdots\!33}{304623237260}a^{23}-\frac{25\!\cdots\!17}{304623237260}a^{22}-\frac{24\!\cdots\!57}{304623237260}a^{21}+\frac{94\!\cdots\!67}{609246474520}a^{20}+\frac{93\!\cdots\!51}{609246474520}a^{19}-\frac{17\!\cdots\!99}{609246474520}a^{18}-\frac{17\!\cdots\!29}{609246474520}a^{17}+\frac{33\!\cdots\!87}{609246474520}a^{16}+\frac{33\!\cdots\!71}{609246474520}a^{15}-\frac{63\!\cdots\!59}{609246474520}a^{14}-\frac{29\!\cdots\!57}{304623237260}a^{13}+\frac{56\!\cdots\!91}{304623237260}a^{12}+\frac{44\!\cdots\!53}{304623237260}a^{11}-\frac{21\!\cdots\!13}{76155809315}a^{10}+\frac{11\!\cdots\!33}{304623237260}a^{9}-\frac{33\!\cdots\!49}{304623237260}a^{8}-\frac{217121378667094}{76155809315}a^{7}+\frac{993266683314159}{1218492949040}a^{6}+\frac{52120754220879}{243698589808}a^{5}-\frac{74511388379209}{1218492949040}a^{4}-\frac{19554011123831}{1218492949040}a^{3}+\frac{5590416065251}{1218492949040}a^{2}-\frac{2973468249163}{1218492949040}a+\frac{1131236066377}{1218492949040}$, $\frac{28076250627}{152311618630}a^{35}+\frac{125923893313}{1218492949040}a^{34}-\frac{1996440101791}{1218492949040}a^{33}-\frac{56648444397}{60924647452}a^{32}+\frac{3705887534579}{304623237260}a^{31}+\frac{2108578842581}{304623237260}a^{30}-\frac{52843105926897}{609246474520}a^{29}-\frac{60173177642543}{1218492949040}a^{28}+\frac{186410658893103}{304623237260}a^{27}+\frac{106152786656743}{304623237260}a^{26}-\frac{327875018983638}{76155809315}a^{25}-\frac{373438768667927}{152311618630}a^{24}+\frac{92\!\cdots\!63}{304623237260}a^{23}+\frac{26\!\cdots\!09}{152311618630}a^{22}-\frac{17\!\cdots\!37}{304623237260}a^{21}-\frac{20\!\cdots\!83}{609246474520}a^{20}+\frac{65\!\cdots\!11}{609246474520}a^{19}+\frac{19\!\cdots\!43}{304623237260}a^{18}-\frac{61\!\cdots\!77}{304623237260}a^{17}-\frac{91\!\cdots\!56}{76155809315}a^{16}+\frac{11\!\cdots\!23}{304623237260}a^{15}+\frac{13\!\cdots\!61}{609246474520}a^{14}-\frac{20\!\cdots\!37}{304623237260}a^{13}-\frac{12\!\cdots\!89}{304623237260}a^{12}+\frac{76\!\cdots\!77}{76155809315}a^{11}+\frac{92\!\cdots\!29}{152311618630}a^{10}+\frac{12\!\cdots\!03}{304623237260}a^{9}-\frac{69\!\cdots\!07}{152311618630}a^{8}-\frac{906627318345251}{304623237260}a^{7}+\frac{25160050443009}{4336273840}a^{6}+\frac{13\!\cdots\!73}{1218492949040}a^{5}-\frac{24719199510611}{76155809315}a^{4}-\frac{9704023628164}{76155809315}a^{3}+\frac{292627391447}{304623237260}a^{2}+\frac{3832009725503}{609246474520}a+\frac{1008325284177}{1218492949040}$, $\frac{130317217}{1218492949040}a^{35}-\frac{16909911449}{243698589808}a^{34}+\frac{49161}{76155809315}a^{33}+\frac{760951008839}{1218492949040}a^{32}-\frac{11911}{304623237260}a^{31}-\frac{1132964067083}{243698589808}a^{30}+\frac{11911}{152311618630}a^{29}+\frac{8082937666689}{243698589808}a^{28}-\frac{14259282827931}{60924647452}a^{26}+\frac{100326504626917}{60924647452}a^{24}-\frac{705329316449239}{60924647452}a^{22}+\frac{11040722212283}{121849294904}a^{21}+\frac{27\!\cdots\!99}{121849294904}a^{20}+\frac{33320138003}{60924647452}a^{19}-\frac{52\!\cdots\!89}{121849294904}a^{18}-\frac{2018324459}{60924647452}a^{17}+\frac{98\!\cdots\!89}{121849294904}a^{16}+\frac{2018324459}{30462323726}a^{15}-\frac{18\!\cdots\!97}{121849294904}a^{14}+\frac{16\!\cdots\!85}{60924647452}a^{12}-\frac{24\!\cdots\!59}{60924647452}a^{10}+\frac{18\!\cdots\!97}{60924647452}a^{8}+\frac{97\!\cdots\!73}{1218492949040}a^{7}-\frac{561967087184617}{243698589808}a^{6}+\frac{14826268512431}{304623237260}a^{5}+\frac{584533702121431}{1218492949040}a^{4}-\frac{278051973476}{76155809315}a^{3}-\frac{3405852030771}{243698589808}a^{2}+\frac{479948137637}{76155809315}a+\frac{255020062429}{243698589808}$, $\frac{51425786428}{76155809315}a^{35}-\frac{33048324019}{304623237260}a^{34}-\frac{7360931256839}{1218492949040}a^{33}+\frac{1180431318753}{1218492949040}a^{32}+\frac{6841197811073}{152311618630}a^{31}-\frac{8772935635543}{1218492949040}a^{30}-\frac{195168110686367}{609246474520}a^{29}+\frac{62562647955271}{1218492949040}a^{28}+\frac{344271594170983}{152311618630}a^{27}-\frac{27588947328883}{76155809315}a^{26}-\frac{48\!\cdots\!43}{304623237260}a^{25}+\frac{776426069933317}{304623237260}a^{24}+\frac{17\!\cdots\!93}{152311618630}a^{23}-\frac{27\!\cdots\!01}{152311618630}a^{22}-\frac{32\!\cdots\!39}{152311618630}a^{21}+\frac{10\!\cdots\!77}{304623237260}a^{20}+\frac{24\!\cdots\!57}{609246474520}a^{19}-\frac{39\!\cdots\!59}{609246474520}a^{18}-\frac{23\!\cdots\!33}{304623237260}a^{17}+\frac{74\!\cdots\!49}{609246474520}a^{16}+\frac{10\!\cdots\!84}{76155809315}a^{15}-\frac{13\!\cdots\!29}{609246474520}a^{14}-\frac{39\!\cdots\!89}{152311618630}a^{13}+\frac{31\!\cdots\!23}{76155809315}a^{12}+\frac{11\!\cdots\!61}{304623237260}a^{11}-\frac{18\!\cdots\!07}{304623237260}a^{10}-\frac{13\!\cdots\!29}{152311618630}a^{9}+\frac{350119464895431}{152311618630}a^{8}+\frac{14\!\cdots\!47}{152311618630}a^{7}+\frac{54915794089828}{76155809315}a^{6}-\frac{19\!\cdots\!03}{1218492949040}a^{5}+\frac{62792909559261}{243698589808}a^{4}+\frac{3336398562893}{60924647452}a^{3}+\frac{3012534464669}{243698589808}a^{2}-\frac{7549722816009}{609246474520}a+\frac{2855710374967}{1218492949040}$
| 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: | \( 695961018492374.6 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{18}\cdot 695961018492374.6 \cdot 1404}{28\cdot\sqrt{840739592110096304569466677876458557457237523263143378686050304}}\cr\approx \mathstrut & 0.280350201158285 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*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^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*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^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*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^36 - 9*x^34 + 67*x^32 - 478*x^30 + 3373*x^28 - 23732*x^26 + 166844*x^24 - 325502*x^22 + 617434*x^20 - 1166722*x^18 + 2181924*x^16 - 3920642*x^14 + 5905564*x^12 - 443012*x^10 + 33233*x^8 - 2493*x^6 + 187*x^4 - 14*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
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 | ${\href{/padicField/3.6.0.1}{6} }^{6}$ | ${\href{/padicField/5.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{6}$ | R | ${\href{/padicField/17.6.0.1}{6} }^{6}$ | ${\href{/padicField/19.6.0.1}{6} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{6}$ | ${\href{/padicField/29.3.0.1}{3} }^{12}$ | ${\href{/padicField/31.6.0.1}{6} }^{6}$ | ${\href{/padicField/37.3.0.1}{3} }^{12}$ | ${\href{/padicField/41.6.0.1}{6} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{6}$ | ${\href{/padicField/53.3.0.1}{3} }^{12}$ | ${\href{/padicField/59.6.0.1}{6} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
|
\(7\)
| Deg $36$ | $6$ | $6$ | $30$ | |||
|
\(13\)
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ | |
| 13.6.4.3 | $x^{6} + 36 x^{5} + 438 x^{4} + 1898 x^{3} + 1344 x^{2} + 5604 x + 21705$ | $3$ | $2$ | $4$ | $C_6$ | $[\ ]_{3}^{2}$ |