Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 8*x^30 + 30*x^28 + 74*x^26 + 143*x^24 + 226*x^22 + 207*x^20 - 185*x^18 - 923*x^16 - 740*x^14 + 3312*x^12 + 14464*x^10 + 36608*x^8 + 75776*x^6 + 122880*x^4 + 131072*x^2 + 65536)
gp: K = bnfinit(y^32 + 8*y^30 + 30*y^28 + 74*y^26 + 143*y^24 + 226*y^22 + 207*y^20 - 185*y^18 - 923*y^16 - 740*y^14 + 3312*y^12 + 14464*y^10 + 36608*y^8 + 75776*y^6 + 122880*y^4 + 131072*y^2 + 65536, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 8*x^30 + 30*x^28 + 74*x^26 + 143*x^24 + 226*x^22 + 207*x^20 - 185*x^18 - 923*x^16 - 740*x^14 + 3312*x^12 + 14464*x^10 + 36608*x^8 + 75776*x^6 + 122880*x^4 + 131072*x^2 + 65536);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 + 8*x^30 + 30*x^28 + 74*x^26 + 143*x^24 + 226*x^22 + 207*x^20 - 185*x^18 - 923*x^16 - 740*x^14 + 3312*x^12 + 14464*x^10 + 36608*x^8 + 75776*x^6 + 122880*x^4 + 131072*x^2 + 65536)
\( x^{32} + 8 x^{30} + 30 x^{28} + 74 x^{26} + 143 x^{24} + 226 x^{22} + 207 x^{20} - 185 x^{18} + \cdots + 65536 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $32$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[0, 16]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(17088578670250951249956505397284700160000000000000000\)
\(\medspace = 2^{32}\cdot 3^{16}\cdot 5^{16}\cdot 29^{8}\cdot 1049^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(42.88\) | 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: | $2\cdot 3^{1/2}5^{1/2}29^{1/2}1049^{1/2}\approx 1351.0218355008183$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(29\), \(1049\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{32768}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{19}-\frac{1}{4}a^{15}+\frac{1}{4}a^{13}-\frac{1}{8}a^{11}+\frac{1}{4}a^{9}-\frac{1}{8}a^{7}-\frac{1}{8}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{20}-\frac{1}{8}a^{16}-\frac{3}{8}a^{14}-\frac{1}{16}a^{12}-\frac{3}{8}a^{10}-\frac{1}{16}a^{8}-\frac{1}{16}a^{6}-\frac{3}{16}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{32}a^{21}-\frac{1}{16}a^{17}-\frac{3}{16}a^{15}-\frac{1}{32}a^{13}+\frac{5}{16}a^{11}-\frac{1}{32}a^{9}+\frac{15}{32}a^{7}+\frac{13}{32}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{64}a^{22}-\frac{1}{32}a^{18}+\frac{13}{32}a^{16}-\frac{1}{64}a^{14}-\frac{11}{32}a^{12}+\frac{31}{64}a^{10}+\frac{15}{64}a^{8}+\frac{13}{64}a^{6}+\frac{5}{16}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{128}a^{23}-\frac{1}{64}a^{19}+\frac{13}{64}a^{17}-\frac{1}{128}a^{15}+\frac{21}{64}a^{13}+\frac{31}{128}a^{11}-\frac{49}{128}a^{9}+\frac{13}{128}a^{7}+\frac{5}{32}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{2816}a^{24}-\frac{3}{128}a^{20}+\frac{109}{1408}a^{18}-\frac{35}{256}a^{16}-\frac{33}{128}a^{14}-\frac{1313}{2816}a^{12}-\frac{115}{256}a^{10}+\frac{7}{256}a^{8}-\frac{219}{704}a^{6}-\frac{3}{16}a^{4}+\frac{5}{11}$, $\frac{1}{5632}a^{25}-\frac{3}{256}a^{21}+\frac{109}{2816}a^{19}-\frac{35}{512}a^{17}+\frac{95}{256}a^{15}-\frac{1313}{5632}a^{13}+\frac{141}{512}a^{11}-\frac{249}{512}a^{9}-\frac{219}{1408}a^{7}+\frac{13}{32}a^{5}+\frac{5}{22}a$, $\frac{1}{11264}a^{26}-\frac{3}{512}a^{22}+\frac{109}{5632}a^{20}-\frac{35}{1024}a^{18}-\frac{161}{512}a^{16}+\frac{4319}{11264}a^{14}+\frac{141}{1024}a^{12}-\frac{249}{1024}a^{10}+\frac{1189}{2816}a^{8}+\frac{13}{64}a^{6}-\frac{1}{2}a^{4}-\frac{17}{44}a^{2}$, $\frac{1}{22528}a^{27}-\frac{3}{1024}a^{23}+\frac{109}{11264}a^{21}-\frac{35}{2048}a^{19}-\frac{161}{1024}a^{17}+\frac{4319}{22528}a^{15}-\frac{883}{2048}a^{13}-\frac{249}{2048}a^{11}+\frac{1189}{5632}a^{9}+\frac{13}{128}a^{7}+\frac{1}{4}a^{5}-\frac{17}{88}a^{3}-\frac{1}{2}a$, $\frac{1}{33746944}a^{28}-\frac{255}{8436736}a^{26}-\frac{215}{2410496}a^{24}-\frac{98583}{16873472}a^{22}+\frac{19431}{33746944}a^{20}+\frac{1411219}{16873472}a^{18}+\frac{3494663}{33746944}a^{16}-\frac{14961013}{33746944}a^{14}+\frac{6884873}{33746944}a^{12}-\frac{789783}{4218368}a^{10}+\frac{184211}{527296}a^{8}+\frac{241211}{527296}a^{6}-\frac{4391}{9416}a^{4}+\frac{4159}{32956}a^{2}+\frac{4}{8239}$, $\frac{1}{67493888}a^{29}-\frac{255}{16873472}a^{27}-\frac{215}{4820992}a^{25}-\frac{98583}{33746944}a^{23}+\frac{19431}{67493888}a^{21}+\frac{1411219}{33746944}a^{19}+\frac{3494663}{67493888}a^{17}-\frac{14961013}{67493888}a^{15}+\frac{6884873}{67493888}a^{13}-\frac{789783}{8436736}a^{11}+\frac{184211}{1054592}a^{9}-\frac{286085}{1054592}a^{7}-\frac{4391}{18832}a^{5}-\frac{28797}{65912}a^{3}-\frac{8235}{16478}a$, $\frac{1}{18763300864}a^{30}-\frac{67}{4690825216}a^{28}+\frac{178223}{9381650432}a^{26}-\frac{725}{852877312}a^{24}-\frac{72846073}{18763300864}a^{22}-\frac{7804307}{1340235776}a^{20}-\frac{1482725945}{18763300864}a^{18}-\frac{2862416101}{18763300864}a^{16}-\frac{1908045191}{18763300864}a^{14}+\frac{364941043}{2345412608}a^{12}-\frac{115426669}{586353152}a^{10}-\frac{236311}{4580884}a^{8}+\frac{771943}{2290442}a^{6}+\frac{3702493}{9161768}a^{4}+\frac{73387}{1145221}a^{2}-\frac{244920}{1145221}$, $\frac{1}{37526601728}a^{31}-\frac{67}{9381650432}a^{29}+\frac{178223}{18763300864}a^{27}-\frac{725}{1705754624}a^{25}-\frac{72846073}{37526601728}a^{23}-\frac{7804307}{2680471552}a^{21}-\frac{1482725945}{37526601728}a^{19}-\frac{2862416101}{37526601728}a^{17}-\frac{1908045191}{37526601728}a^{15}-\frac{1980471565}{4690825216}a^{13}+\frac{470926483}{1172706304}a^{11}+\frac{4344573}{9161768}a^{9}-\frac{1518499}{4580884}a^{7}-\frac{5459275}{18323536}a^{5}-\frac{535917}{1145221}a^{3}+\frac{900301}{2290442}a$
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
$C_{30}$, which has order $30$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $30$ (assuming GRH)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $15$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{1290159}{1172706304} a^{31} - \frac{8677503}{1340235776} a^{29} - \frac{45282717}{2345412608} a^{27} - \frac{17390893}{426438656} a^{25} - \frac{48202419}{670117888} a^{23} - \frac{920999199}{9381650432} a^{21} - \frac{92951135}{4690825216} a^{19} + \frac{2355067401}{9381650432} a^{17} + \frac{4626464757}{9381650432} a^{15} - \frac{1978207937}{9381650432} a^{13} - \frac{3730563861}{1172706304} a^{11} - \frac{386325627}{41882368} a^{9} - \frac{3075879777}{146588288} a^{7} - \frac{361367121}{9161768} a^{5} - \frac{34097907}{654412} a^{3} - \frac{38542751}{1145221} a \)
(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: |
$\frac{30520821}{18763300864}a^{30}+\frac{2822483}{293176576}a^{28}+\frac{24317801}{852877312}a^{26}+\frac{559019345}{9381650432}a^{24}+\frac{1955246507}{18763300864}a^{22}+\frac{120126619}{852877312}a^{20}+\frac{70134445}{2680471552}a^{18}-\frac{989735315}{2680471552}a^{16}-\frac{1219425565}{1705754624}a^{14}+\frac{153067559}{426438656}a^{12}+\frac{2772293677}{586353152}a^{10}+\frac{360981455}{26652416}a^{8}+\frac{1117793407}{36647072}a^{6}+\frac{1046555995}{18323536}a^{4}+\frac{7861167}{104111}a^{2}+\frac{55089939}{1145221}$, $\frac{9398859}{18763300864}a^{30}+\frac{7983863}{2345412608}a^{28}+\frac{98107349}{9381650432}a^{26}+\frac{207388151}{9381650432}a^{24}+\frac{744106213}{18763300864}a^{22}+\frac{524753099}{9381650432}a^{20}+\frac{338773381}{18763300864}a^{18}-\frac{2557878227}{18763300864}a^{16}-\frac{758809295}{2680471552}a^{14}+\frac{43499531}{670117888}a^{12}+\frac{1954929971}{1172706304}a^{10}+\frac{360961239}{73294144}a^{8}+\frac{415516429}{36647072}a^{6}+\frac{402978227}{18323536}a^{4}+\frac{33722727}{1145221}a^{2}+\frac{2042398}{104111}$, $\frac{1290159}{1172706304}a^{31}+\frac{8677503}{1340235776}a^{29}+\frac{45282717}{2345412608}a^{27}+\frac{17390893}{426438656}a^{25}+\frac{48202419}{670117888}a^{23}+\frac{920999199}{9381650432}a^{21}+\frac{92951135}{4690825216}a^{19}-\frac{2355067401}{9381650432}a^{17}-\frac{4626464757}{9381650432}a^{15}+\frac{1978207937}{9381650432}a^{13}+\frac{3730563861}{1172706304}a^{11}+\frac{386325627}{41882368}a^{9}+\frac{3075879777}{146588288}a^{7}+\frac{361367121}{9161768}a^{5}+\frac{34097907}{654412}a^{3}+\frac{38542751}{1145221}a+1$, $\frac{20643363}{37526601728}a^{31}+\frac{4363925}{1340235776}a^{29}+\frac{175505997}{18763300864}a^{27}+\frac{370847003}{18763300864}a^{25}+\frac{187002435}{5360943104}a^{23}+\frac{873667033}{18763300864}a^{21}+\frac{325501269}{37526601728}a^{19}-\frac{4716790143}{37526601728}a^{17}-\frac{8717757093}{37526601728}a^{15}+\frac{610404079}{4690825216}a^{13}+\frac{1864563559}{1172706304}a^{11}+\frac{377587173}{83764736}a^{9}+\frac{374476835}{36647072}a^{7}+\frac{3313917}{171248}a^{5}+\frac{4020833}{163603}a^{3}+\frac{17979872}{1145221}a$, $\frac{7220639}{18763300864}a^{30}+\frac{2840295}{1172706304}a^{28}+\frac{68158105}{9381650432}a^{26}+\frac{143925987}{9381650432}a^{24}+\frac{537143009}{18763300864}a^{22}+\frac{401851651}{9381650432}a^{20}+\frac{320699441}{18763300864}a^{18}-\frac{1717145471}{18763300864}a^{16}-\frac{3710380365}{18763300864}a^{14}+\frac{16366145}{426438656}a^{12}+\frac{11622713}{10470592}a^{10}+\frac{249330055}{73294144}a^{8}+\frac{587029523}{73294144}a^{6}+\frac{146022397}{9161768}a^{4}+\frac{50395085}{2290442}a^{2}+\frac{16308580}{1145221}$, $\frac{14179265}{9381650432}a^{31}-\frac{215137}{18763300864}a^{30}+\frac{86737669}{9381650432}a^{29}-\frac{533825}{1172706304}a^{28}+\frac{18699303}{670117888}a^{27}-\frac{1336317}{852877312}a^{26}+\frac{34639265}{586353152}a^{25}-\frac{38877757}{9381650432}a^{24}+\frac{981021753}{9381650432}a^{23}-\frac{147767615}{18763300864}a^{22}+\frac{1358910797}{9381650432}a^{21}-\frac{10378535}{852877312}a^{20}+\frac{358659085}{9381650432}a^{19}-\frac{1698013}{175357952}a^{18}-\frac{1673989111}{4690825216}a^{17}+\frac{406585057}{18763300864}a^{16}-\frac{1707075523}{2345412608}a^{15}+\frac{91931849}{1705754624}a^{14}+\frac{2351991409}{9381650432}a^{13}+\frac{20787653}{426438656}a^{12}+\frac{5333797005}{1172706304}a^{11}-\frac{265687567}{1172706304}a^{10}+\frac{973319359}{73294144}a^{9}-\frac{5510381}{6663104}a^{8}+\frac{632820709}{20941184}a^{7}-\frac{75205785}{36647072}a^{6}+\frac{2103413985}{36647072}a^{5}-\frac{79610607}{18323536}a^{4}+\frac{353572767}{4580884}a^{3}-\frac{2382511}{416444}a^{2}+\frac{8379190}{163603}a-\frac{7222927}{1145221}$, $\frac{15064513}{18763300864}a^{31}+\frac{9783}{243679232}a^{30}+\frac{21565013}{4690825216}a^{29}-\frac{55449}{213219328}a^{28}+\frac{18173789}{1340235776}a^{27}-\frac{13577243}{9381650432}a^{26}+\frac{269180985}{9381650432}a^{25}-\frac{5420319}{1340235776}a^{24}+\frac{944985879}{18763300864}a^{23}-\frac{12135185}{1705754624}a^{22}+\frac{639793507}{9381650432}a^{21}-\frac{130502893}{9381650432}a^{20}+\frac{21114893}{1705754624}a^{19}-\frac{337168891}{18763300864}a^{18}-\frac{3344724181}{18763300864}a^{17}+\frac{18318551}{1705754624}a^{16}-\frac{6402180319}{18763300864}a^{15}+\frac{1199477703}{18763300864}a^{14}+\frac{52540927}{293176576}a^{13}+\frac{441287185}{4690825216}a^{12}+\frac{5278228807}{2345412608}a^{11}-\frac{11628665}{106609664}a^{10}+\frac{3796917711}{586353152}a^{9}-\frac{220613049}{293176576}a^{8}+\frac{155254811}{10470592}a^{7}-\frac{32118771}{18323536}a^{6}+\frac{505794571}{18323536}a^{5}-\frac{985111}{237968}a^{4}+\frac{330329239}{9161768}a^{3}-\frac{33294697}{4580884}a^{2}+\frac{7497617}{327206}a-\frac{9649480}{1145221}$, $\frac{16841973}{37526601728}a^{31}+\frac{13720537}{9381650432}a^{30}+\frac{11861543}{4690825216}a^{29}+\frac{19942851}{2345412608}a^{28}+\frac{136641147}{18763300864}a^{27}+\frac{118053323}{4690825216}a^{26}+\frac{26099307}{1705754624}a^{25}+\frac{246224633}{4690825216}a^{24}+\frac{993752315}{37526601728}a^{23}+\frac{853657039}{9381650432}a^{22}+\frac{661458205}{18763300864}a^{21}+\frac{576657647}{4690825216}a^{20}+\frac{168145531}{37526601728}a^{19}+\frac{183980791}{9381650432}a^{18}-\frac{3698015133}{37526601728}a^{17}-\frac{3064026309}{9381650432}a^{16}-\frac{6548921639}{37526601728}a^{15}-\frac{5859268879}{9381650432}a^{14}+\frac{1006267543}{9381650432}a^{13}+\frac{398135641}{1172706304}a^{12}+\frac{419650621}{335058944}a^{11}+\frac{4954690817}{1172706304}a^{10}+\frac{255320057}{73294144}a^{9}+\frac{3507765193}{293176576}a^{8}+\frac{1155606505}{146588288}a^{7}+\frac{1972539979}{73294144}a^{6}+\frac{134244833}{9161768}a^{5}+\frac{919456247}{18323536}a^{4}+\frac{86609825}{4580884}a^{3}+\frac{150906047}{2290442}a^{2}+\frac{13043166}{1145221}a+\frac{47492325}{1145221}$, $\frac{21831101}{37526601728}a^{31}-\frac{80485}{852877312}a^{30}+\frac{1248221}{335058944}a^{29}-\frac{2576029}{4690825216}a^{28}+\frac{215230243}{18763300864}a^{27}-\frac{8928097}{4690825216}a^{26}+\frac{459961065}{18763300864}a^{25}-\frac{20723585}{4690825216}a^{24}+\frac{233872261}{5360943104}a^{23}-\frac{82664749}{9381650432}a^{22}+\frac{1158329321}{18763300864}a^{21}-\frac{31292875}{2345412608}a^{20}+\frac{772933123}{37526601728}a^{19}-\frac{3576059}{1340235776}a^{18}-\frac{5432469725}{37526601728}a^{17}+\frac{38370135}{1340235776}a^{16}-\frac{11534630743}{37526601728}a^{15}+\frac{644598751}{9381650432}a^{14}+\frac{612682685}{9381650432}a^{13}+\frac{102736041}{4690825216}a^{12}+\frac{4274500635}{2345412608}a^{11}-\frac{329232821}{1172706304}a^{10}+\frac{461656599}{83764736}a^{9}-\frac{71768143}{73294144}a^{8}+\frac{917750861}{73294144}a^{7}-\frac{180606409}{73294144}a^{6}+\frac{8234673}{342496}a^{5}-\frac{44024823}{9161768}a^{4}+\frac{42841575}{1308824}a^{3}-\frac{13845605}{2290442}a^{2}+\frac{24737652}{1145221}a-\frac{3226292}{1145221}$, $\frac{21058133}{37526601728}a^{31}-\frac{75173345}{18763300864}a^{30}+\frac{15412035}{4690825216}a^{29}-\frac{114080091}{4690825216}a^{28}+\frac{180950755}{18763300864}a^{27}-\frac{687035271}{9381650432}a^{26}+\frac{375844921}{18763300864}a^{25}-\frac{1456363681}{9381650432}a^{24}+\frac{1323237691}{37526601728}a^{23}-\frac{5134855287}{18763300864}a^{22}+\frac{875950237}{18763300864}a^{21}-\frac{508385465}{1340235776}a^{20}+\frac{247727819}{37526601728}a^{19}-\frac{17574325}{175357952}a^{18}-\frac{4767283069}{37526601728}a^{17}+\frac{17507717533}{18763300864}a^{16}-\frac{1282103633}{5360943104}a^{15}+\frac{35555144463}{18763300864}a^{14}+\frac{181688845}{1340235776}a^{13}-\frac{2245109}{3331552}a^{12}+\frac{958012149}{586353152}a^{11}-\frac{13967440315}{1172706304}a^{10}+\frac{1342922699}{293176576}a^{9}-\frac{319443615}{9161768}a^{8}+\frac{1505821447}{146588288}a^{7}-\frac{1454799221}{18323536}a^{6}+\frac{704788523}{36647072}a^{5}-\frac{689011075}{4580884}a^{4}+\frac{114557879}{4580884}a^{3}-\frac{231927316}{1145221}a^{2}+\frac{16438229}{1145221}a-\frac{156000970}{1145221}$, $\frac{3070939}{37526601728}a^{31}-\frac{4922969}{18763300864}a^{30}+\frac{218661}{426438656}a^{29}-\frac{8618571}{4690825216}a^{28}+\frac{3875899}{2680471552}a^{27}-\frac{60434431}{9381650432}a^{26}+\frac{46379367}{18763300864}a^{25}-\frac{134714457}{9381650432}a^{24}+\frac{15065343}{3411509248}a^{23}-\frac{497835199}{18763300864}a^{22}+\frac{85082059}{18763300864}a^{21}-\frac{406314663}{9381650432}a^{20}-\frac{157478459}{37526601728}a^{19}-\frac{514280015}{18763300864}a^{18}-\frac{77945641}{3411509248}a^{17}+\frac{1289897557}{18763300864}a^{16}-\frac{1368764809}{37526601728}a^{15}+\frac{559078769}{2680471552}a^{14}+\frac{477420257}{9381650432}a^{13}+\frac{1673703}{20941184}a^{12}+\frac{29883341}{106609664}a^{11}-\frac{1025270493}{1172706304}a^{10}+\frac{96116249}{146588288}a^{9}-\frac{887456705}{293176576}a^{8}+\frac{2421407}{1903744}a^{7}-\frac{134481283}{18323536}a^{6}+\frac{7551167}{3331552}a^{5}-\frac{68114715}{4580884}a^{4}+\frac{10769739}{4580884}a^{3}-\frac{105930807}{4580884}a^{2}-\frac{5800}{163603}a-\frac{19828894}{1145221}$, $\frac{360253}{350715904}a^{31}+\frac{3048157}{2680471552}a^{30}+\frac{28696699}{4690825216}a^{29}+\frac{15663651}{2345412608}a^{28}+\frac{347708561}{18763300864}a^{27}+\frac{184087949}{9381650432}a^{26}+\frac{730670459}{18763300864}a^{25}+\frac{55786993}{1340235776}a^{24}+\frac{2577554737}{37526601728}a^{23}+\frac{1358907909}{18763300864}a^{22}+\frac{1756657503}{18763300864}a^{21}+\frac{910803595}{9381650432}a^{20}+\frac{745711921}{37526601728}a^{19}+\frac{333855125}{18763300864}a^{18}-\frac{8980250919}{37526601728}a^{17}-\frac{4776854035}{18763300864}a^{16}-\frac{17792107525}{37526601728}a^{15}-\frac{9113266489}{18763300864}a^{14}+\frac{1877519297}{9381650432}a^{13}+\frac{1150582249}{4690825216}a^{12}+\frac{1013515163}{335058944}a^{11}+\frac{1927672475}{586353152}a^{10}+\frac{2601067905}{293176576}a^{9}+\frac{2769407813}{293176576}a^{8}+\frac{2928561389}{146588288}a^{7}+\frac{777482763}{36647072}a^{6}+\frac{1370793987}{36647072}a^{5}+\frac{103312625}{2617648}a^{4}+\frac{456030623}{9161768}a^{3}+\frac{237208061}{4580884}a^{2}+\frac{73080207}{2290442}a+\frac{3622408}{104111}$, $\frac{1093919}{9381650432}a^{31}-\frac{1557821}{18763300864}a^{30}+\frac{9468153}{9381650432}a^{29}+\frac{122429}{1172706304}a^{28}+\frac{226209}{60919808}a^{27}+\frac{5947861}{9381650432}a^{26}+\frac{19708401}{2345412608}a^{25}+\frac{18070967}{9381650432}a^{24}+\frac{133202747}{9381650432}a^{23}+\frac{106151165}{18763300864}a^{22}+\frac{17123175}{852877312}a^{21}+\frac{17225921}{1340235776}a^{20}+\frac{8576457}{852877312}a^{19}+\frac{27253063}{1705754624}a^{18}-\frac{46635117}{1172706304}a^{17}-\frac{168790019}{18763300864}a^{16}-\frac{49645461}{426438656}a^{15}-\frac{918650393}{18763300864}a^{14}-\frac{286153239}{9381650432}a^{13}-\frac{445837417}{4690825216}a^{12}+\frac{1294503355}{2345412608}a^{11}-\frac{4969031}{293176576}a^{10}+\frac{48341299}{26652416}a^{9}+\frac{89086783}{293176576}a^{8}+\frac{84893995}{20941184}a^{7}+\frac{77754539}{73294144}a^{6}+\frac{138778271}{18323536}a^{5}+\frac{30396279}{9161768}a^{4}+\frac{4692263}{416444}a^{3}+\frac{27790535}{4580884}a^{2}+\frac{1346956}{163603}a+\frac{5907457}{1145221}$, $\frac{6483061}{37526601728}a^{31}+\frac{4945803}{4690825216}a^{30}+\frac{800159}{1172706304}a^{29}+\frac{15024591}{2345412608}a^{28}+\frac{37543923}{18763300864}a^{27}+\frac{44634973}{2345412608}a^{26}+\frac{81010481}{18763300864}a^{25}+\frac{95253161}{2345412608}a^{24}+\frac{272635819}{37526601728}a^{23}+\frac{334309793}{4690825216}a^{22}+\frac{189106425}{18763300864}a^{21}+\frac{28493735}{293176576}a^{20}+\frac{10429645}{5360943104}a^{19}+\frac{101052625}{4690825216}a^{18}-\frac{95366323}{5360943104}a^{17}-\frac{1175136713}{4690825216}a^{16}-\frac{1429347263}{37526601728}a^{15}-\frac{2342169055}{4690825216}a^{14}+\frac{477162141}{9381650432}a^{13}+\frac{40600659}{213219328}a^{12}+\frac{96168201}{293176576}a^{11}+\frac{265404087}{83764736}a^{10}+\frac{549766331}{586353152}a^{9}+\frac{2689350817}{293176576}a^{8}+\frac{317283643}{146588288}a^{7}+\frac{192602281}{9161768}a^{6}+\frac{131632491}{36647072}a^{5}+\frac{724253993}{18323536}a^{4}+\frac{22855733}{4580884}a^{3}+\frac{240788543}{4580884}a^{2}+\frac{7567375}{2290442}a+\frac{40497715}{1145221}$, $\frac{1977443}{18763300864}a^{31}-\frac{6934197}{1172706304}a^{30}+\frac{2012825}{1172706304}a^{29}-\frac{23886105}{670117888}a^{28}+\frac{60205517}{9381650432}a^{27}-\frac{125681085}{1172706304}a^{26}+\frac{149989639}{9381650432}a^{25}-\frac{533101509}{2345412608}a^{24}+\frac{557467901}{18763300864}a^{23}-\frac{134096731}{335058944}a^{22}+\frac{443016215}{9381650432}a^{21}-\frac{2577159217}{4690825216}a^{20}+\frac{692160125}{18763300864}a^{19}-\frac{319348867}{2345412608}a^{18}-\frac{1388967523}{18763300864}a^{17}+\frac{6482212523}{4690825216}a^{16}-\frac{4531309481}{18763300864}a^{15}+\frac{12983789015}{4690825216}a^{14}-\frac{79043127}{426438656}a^{13}-\frac{4938880055}{4690825216}a^{12}+\frac{142512787}{167529472}a^{11}-\frac{10297859735}{586353152}a^{10}+\frac{1929530619}{586353152}a^{9}-\frac{1072610393}{20941184}a^{8}+\frac{287637523}{36647072}a^{7}-\frac{8518736003}{73294144}a^{6}+\frac{295932965}{18323536}a^{5}-\frac{1005655723}{4580884}a^{4}+\frac{110766157}{4580884}a^{3}-\frac{95686081}{327206}a^{2}+\frac{24609614}{1145221}a-\frac{223727047}{1145221}$
| 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: | \( 1780421557670.7527 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 1780421557670.7527 \cdot 30}{12\cdot\sqrt{17088578670250951249956505397284700160000000000000000}}\cr\approx \mathstrut & 0.200903626160088 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 + 8*x^30 + 30*x^28 + 74*x^26 + 143*x^24 + 226*x^22 + 207*x^20 - 185*x^18 - 923*x^16 - 740*x^14 + 3312*x^12 + 14464*x^10 + 36608*x^8 + 75776*x^6 + 122880*x^4 + 131072*x^2 + 65536)
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^32 + 8*x^30 + 30*x^28 + 74*x^26 + 143*x^24 + 226*x^22 + 207*x^20 - 185*x^18 - 923*x^16 - 740*x^14 + 3312*x^12 + 14464*x^10 + 36608*x^8 + 75776*x^6 + 122880*x^4 + 131072*x^2 + 65536, 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^32 + 8*x^30 + 30*x^28 + 74*x^26 + 143*x^24 + 226*x^22 + 207*x^20 - 185*x^18 - 923*x^16 - 740*x^14 + 3312*x^12 + 14464*x^10 + 36608*x^8 + 75776*x^6 + 122880*x^4 + 131072*x^2 + 65536);
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^32 + 8*x^30 + 30*x^28 + 74*x^26 + 143*x^24 + 226*x^22 + 207*x^20 - 185*x^18 - 923*x^16 - 740*x^14 + 3312*x^12 + 14464*x^10 + 36608*x^8 + 75776*x^6 + 122880*x^4 + 131072*x^2 + 65536);
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
$D_4^2:C_2^3$ (as 32T12882):
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)
| A solvable group of order 512 |
| The 80 conjugacy class representatives for $D_4^2:C_2^3$ |
| Character table for $D_4^2:C_2^3$ |
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]
Sibling fields
| Degree 32 siblings: | data not computed |
| Minimal sibling: | not computed |
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 | R | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{8}$ | ${\href{/padicField/11.2.0.1}{2} }^{16}$ | ${\href{/padicField/13.4.0.1}{4} }^{8}$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.2.0.1}{2} }^{16}$ | ${\href{/padicField/37.8.0.1}{8} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{8}$ | ${\href{/padicField/47.8.0.1}{8} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| 2.8.8.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(3\)
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ |
| 3.16.8.1 | $x^{16} + 24 x^{14} + 4 x^{13} + 254 x^{12} + 24 x^{11} + 1508 x^{10} - 172 x^{9} + 5273 x^{8} - 2344 x^{7} + 11640 x^{6} - 7392 x^{5} + 22724 x^{4} - 10768 x^{3} + 19008 x^{2} - 11056 x + 8596$ | $2$ | $8$ | $8$ | $C_8\times C_2$ | $[\ ]_{2}^{8}$ | |
|
\(5\)
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(1049\)
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |