↑

Properties

Label 32.0.126...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.268\times 10^{54}$
Root discriminant \(49.06\)
Ramified primes $2,5$
Class number $85$ (GRH)
Class group [85] (GRH)
Galois group $C_4\times C_8$ (as 32T43)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 8*x^30 + 44*x^28 - 208*x^26 + 910*x^24 - 2800*x^22 + 7440*x^20 - 17664*x^18 + 35492*x^16 - 44096*x^14 + 49952*x^12 - 50240*x^10 + 37432*x^8 - 5696*x^6 + 864*x^4 - 128*x^2 + 16)
 
gp: K = bnfinit(y^32 - 8*y^30 + 44*y^28 - 208*y^26 + 910*y^24 - 2800*y^22 + 7440*y^20 - 17664*y^18 + 35492*y^16 - 44096*y^14 + 49952*y^12 - 50240*y^10 + 37432*y^8 - 5696*y^6 + 864*y^4 - 128*y^2 + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 8*x^30 + 44*x^28 - 208*x^26 + 910*x^24 - 2800*x^22 + 7440*x^20 - 17664*x^18 + 35492*x^16 - 44096*x^14 + 49952*x^12 - 50240*x^10 + 37432*x^8 - 5696*x^6 + 864*x^4 - 128*x^2 + 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 8*x^30 + 44*x^28 - 208*x^26 + 910*x^24 - 2800*x^22 + 7440*x^20 - 17664*x^18 + 35492*x^16 - 44096*x^14 + 49952*x^12 - 50240*x^10 + 37432*x^8 - 5696*x^6 + 864*x^4 - 128*x^2 + 16)
 

\( x^{32} - 8 x^{30} + 44 x^{28} - 208 x^{26} + 910 x^{24} - 2800 x^{22} + 7440 x^{20} - 17664 x^{18} + \cdots + 16 \) Copy content Toggle raw display

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:   \(1267650600228229401496703205376000000000000000000000000\) \(\medspace = 2^{124}\cdot 5^{24}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(49.06\)
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:  $2^{31/8}5^{3/4}\approx 49.05900508138431$
Ramified primes:   \(2\), \(5\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(160=2^{5}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(3,·)$, $\chi_{160}(129,·)$, $\chi_{160}(9,·)$, $\chi_{160}(11,·)$, $\chi_{160}(17,·)$, $\chi_{160}(19,·)$, $\chi_{160}(153,·)$, $\chi_{160}(27,·)$, $\chi_{160}(33,·)$, $\chi_{160}(41,·)$, $\chi_{160}(43,·)$, $\chi_{160}(49,·)$, $\chi_{160}(51,·)$, $\chi_{160}(137,·)$, $\chi_{160}(57,·)$, $\chi_{160}(59,·)$, $\chi_{160}(67,·)$, $\chi_{160}(73,·)$, $\chi_{160}(81,·)$, $\chi_{160}(83,·)$, $\chi_{160}(139,·)$, $\chi_{160}(89,·)$, $\chi_{160}(91,·)$, $\chi_{160}(97,·)$, $\chi_{160}(99,·)$, $\chi_{160}(131,·)$, $\chi_{160}(107,·)$, $\chi_{160}(113,·)$, $\chi_{160}(147,·)$, $\chi_{160}(121,·)$, $\chi_{160}(123,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{4}a^{16}$, $\frac{1}{4}a^{17}$, $\frac{1}{124}a^{18}+\frac{15}{124}a^{16}+\frac{2}{31}a^{14}-\frac{1}{31}a^{12}+\frac{1}{62}a^{10}-\frac{13}{62}a^{8}+\frac{11}{31}a^{6}+\frac{10}{31}a^{4}-\frac{5}{31}a^{2}-\frac{13}{31}$, $\frac{1}{124}a^{19}+\frac{15}{124}a^{17}+\frac{2}{31}a^{15}-\frac{1}{31}a^{13}+\frac{1}{62}a^{11}-\frac{13}{62}a^{9}+\frac{11}{31}a^{7}+\frac{10}{31}a^{5}-\frac{5}{31}a^{3}-\frac{13}{31}a$, $\frac{1}{124}a^{20}+\frac{3}{62}a^{10}+\frac{9}{31}$, $\frac{1}{124}a^{21}+\frac{3}{62}a^{11}+\frac{9}{31}a$, $\frac{1}{124}a^{22}+\frac{3}{62}a^{12}+\frac{9}{31}a^{2}$, $\frac{1}{124}a^{23}+\frac{3}{62}a^{13}+\frac{9}{31}a^{3}$, $\frac{1}{248}a^{24}-\frac{7}{31}a^{14}-\frac{11}{31}a^{4}$, $\frac{1}{248}a^{25}-\frac{7}{31}a^{15}-\frac{11}{31}a^{5}$, $\frac{1}{102362248}a^{26}+\frac{3173}{1651004}a^{24}+\frac{85069}{51181124}a^{22}-\frac{85657}{25590562}a^{20}+\frac{79539}{25590562}a^{18}+\frac{411907}{51181124}a^{16}-\frac{573153}{25590562}a^{14}-\frac{31366}{12795281}a^{12}+\frac{3144819}{25590562}a^{10}+\frac{3841175}{25590562}a^{8}-\frac{5143468}{12795281}a^{6}+\frac{3127171}{12795281}a^{4}-\frac{1006672}{12795281}a^{2}-\frac{4928301}{12795281}$, $\frac{1}{102362248}a^{27}+\frac{3173}{1651004}a^{25}+\frac{85069}{51181124}a^{23}-\frac{85657}{25590562}a^{21}+\frac{79539}{25590562}a^{19}+\frac{411907}{51181124}a^{17}-\frac{573153}{25590562}a^{15}-\frac{31366}{12795281}a^{13}+\frac{3144819}{25590562}a^{11}+\frac{3841175}{25590562}a^{9}-\frac{5143468}{12795281}a^{7}+\frac{3127171}{12795281}a^{5}-\frac{1006672}{12795281}a^{3}-\frac{4928301}{12795281}a$, $\frac{1}{3173229688}a^{28}+\frac{15}{3173229688}a^{26}-\frac{1353545}{3173229688}a^{24}-\frac{2071545}{793307422}a^{22}-\frac{4501075}{1586614844}a^{20}+\frac{1974767}{1586614844}a^{18}+\frac{44767237}{1586614844}a^{16}-\frac{122974903}{793307422}a^{14}+\frac{11094585}{396653711}a^{12}+\frac{181713821}{793307422}a^{10}-\frac{35684806}{396653711}a^{8}+\frac{174213533}{396653711}a^{6}-\frac{90689607}{396653711}a^{4}+\frac{125886043}{396653711}a^{2}+\frac{9949030}{396653711}$, $\frac{1}{3173229688}a^{29}+\frac{15}{3173229688}a^{27}-\frac{1353545}{3173229688}a^{25}-\frac{2071545}{793307422}a^{23}-\frac{4501075}{1586614844}a^{21}+\frac{1974767}{1586614844}a^{19}+\frac{44767237}{1586614844}a^{17}-\frac{122974903}{793307422}a^{15}+\frac{11094585}{396653711}a^{13}+\frac{181713821}{793307422}a^{11}-\frac{35684806}{396653711}a^{9}+\frac{174213533}{396653711}a^{7}-\frac{90689607}{396653711}a^{5}+\frac{125886043}{396653711}a^{3}+\frac{9949030}{396653711}a$, $\frac{1}{3173229688}a^{30}+\frac{57208}{396653711}a^{20}-\frac{54751068}{396653711}a^{10}-\frac{130294233}{396653711}$, $\frac{1}{3173229688}a^{31}+\frac{57208}{396653711}a^{21}-\frac{54751068}{396653711}a^{11}-\frac{130294233}{396653711}a$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{85}$, which has order $85$ (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:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{861250}{396653711} a^{30} + \frac{6201000}{396653711} a^{28} - \frac{32469125}{396653711} a^{26} + \frac{1196091895}{3173229688} a^{24} - \frac{644215000}{396653711} a^{22} + \frac{1802424000}{396653711} a^{20} - \frac{4556873750}{396653711} a^{18} + \frac{10328110000}{396653711} a^{16} - \frac{19039316666}{396653711} a^{14} + \frac{15045004000}{396653711} a^{12} - \frac{15695764500}{396653711} a^{10} + \frac{12650040000}{396653711} a^{8} - \frac{1925066000}{396653711} a^{6} - \frac{16822240718}{396653711} a^{4} - \frac{43407000}{396653711} a^{2} + \frac{5512000}{396653711} \)  (order $10$) Copy content Toggle raw display
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:   $\frac{93249}{793307422}a^{30}-\frac{1025739}{1586614844}a^{28}+\frac{1212237}{396653711}a^{26}-\frac{42428295}{3173229688}a^{24}+\frac{22194523}{396653711}a^{22}-\frac{1398735}{12795281}a^{20}+\frac{102946896}{396653711}a^{18}-\frac{827398377}{1586614844}a^{16}+\frac{256994244}{396653711}a^{14}+\frac{521935826}{396653711}a^{12}+\frac{292801860}{396653711}a^{10}-\frac{436312071}{793307422}a^{8}+\frac{33196644}{396653711}a^{6}-\frac{5035446}{396653711}a^{4}+\frac{2606246504}{396653711}a^{2}-\frac{396746960}{396653711}$, $\frac{18224637}{793307422}a^{30}-\frac{583195333}{3173229688}a^{28}+\frac{400942014}{396653711}a^{26}-\frac{1895362248}{396653711}a^{24}+\frac{8292209835}{396653711}a^{22}-\frac{25514491800}{396653711}a^{20}+\frac{67794764959}{396653711}a^{18}-\frac{160959993984}{396653711}a^{16}+\frac{323414408202}{396653711}a^{14}-\frac{401816796576}{396653711}a^{12}+\frac{455178533712}{396653711}a^{10}-\frac{915853891225}{793307422}a^{8}+\frac{341092306092}{396653711}a^{6}-\frac{51903766176}{396653711}a^{4}+\frac{7873043184}{396653711}a^{2}-\frac{1166376768}{396653711}$, $\frac{449209}{793307422}a^{30}-\frac{4941299}{1586614844}a^{28}+\frac{5839717}{396653711}a^{26}-\frac{204390095}{3173229688}a^{24}+\frac{106914249}{396653711}a^{22}-\frac{6738135}{12795281}a^{20}+\frac{495926736}{396653711}a^{18}-\frac{3985831457}{1586614844}a^{16}+\frac{1238020004}{396653711}a^{14}+\frac{5020239691}{793307422}a^{12}+\frac{1410516260}{396653711}a^{10}-\frac{2101848911}{793307422}a^{8}+\frac{159918404}{396653711}a^{6}-\frac{24257286}{396653711}a^{4}+\frac{11108202117}{396653711}a^{2}-\frac{449209}{396653711}$, $\frac{93249}{793307422}a^{30}-\frac{1025739}{1586614844}a^{28}+\frac{1212237}{396653711}a^{26}-\frac{42428295}{3173229688}a^{24}+\frac{22194523}{396653711}a^{22}-\frac{1398735}{12795281}a^{20}+\frac{102946896}{396653711}a^{18}-\frac{827398377}{1586614844}a^{16}+\frac{256994244}{396653711}a^{14}+\frac{521935826}{396653711}a^{12}+\frac{292801860}{396653711}a^{10}-\frac{436312071}{793307422}a^{8}+\frac{33196644}{396653711}a^{6}-\frac{5035446}{396653711}a^{4}+\frac{2209592793}{396653711}a^{2}-\frac{93249}{396653711}$, $\frac{5034295}{793307422}a^{30}-\frac{84564477}{1586614844}a^{28}+\frac{237337793}{793307422}a^{26}-\frac{2267119337}{1586614844}a^{24}+\frac{2494117535}{396653711}a^{22}-\frac{7936436536}{396653711}a^{20}+\frac{21428346461}{396653711}a^{18}-\frac{51588371800}{396653711}a^{16}+\frac{106158527310}{396653711}a^{14}-\frac{144446689296}{396653711}a^{12}+\frac{165594508058}{396653711}a^{10}-\frac{341942244967}{793307422}a^{8}+\frac{138438021204}{396653711}a^{6}-\frac{45270584043}{396653711}a^{4}+\frac{3196904508}{396653711}a^{2}-\frac{78301777}{396653711}$, $\frac{16338405}{793307422}a^{30}-\frac{128443473}{793307422}a^{28}+\frac{2803849767}{3173229688}a^{26}-\frac{13201004655}{3173229688}a^{24}+\frac{7202553475}{396653711}a^{22}-\frac{228928125}{4153442}a^{20}+\frac{57692024916}{396653711}a^{18}-\frac{272263747455}{793307422}a^{16}+\frac{270608880396}{396653711}a^{14}-\frac{321615446365}{396653711}a^{12}+\frac{361260168180}{396653711}a^{10}-\frac{715158195447}{793307422}a^{8}+\frac{252878246575}{396653711}a^{6}-\frac{8265697272}{396653711}a^{4}+\frac{3840878360}{396653711}a^{2}-\frac{559198520}{396653711}$, $\frac{1815749}{793307422}a^{30}-\frac{25829739}{1586614844}a^{28}+\frac{33681362}{396653711}a^{26}-\frac{619260095}{1586614844}a^{24}+\frac{666409523}{396653711}a^{22}-\frac{1845784785}{396653711}a^{20}+\frac{4659820646}{396653711}a^{18}-\frac{42139838377}{1586614844}a^{16}+\frac{19296310910}{396653711}a^{14}-\frac{14523068174}{396653711}a^{12}+\frac{15988566360}{396653711}a^{10}-\frac{25736392071}{793307422}a^{8}+\frac{10252684}{2076721}a^{6}+\frac{16817205272}{396653711}a^{4}+\frac{2252999793}{396653711}a^{2}+\frac{391048462}{396653711}$, $\frac{93249}{793307422}a^{31}-\frac{4588028}{396653711}a^{30}-\frac{1025739}{1586614844}a^{29}+\frac{72261441}{793307422}a^{28}+\frac{1212237}{396653711}a^{27}-\frac{197286598}{396653711}a^{26}-\frac{42428295}{3173229688}a^{25}+\frac{929075670}{396653711}a^{24}+\frac{22194523}{396653711}a^{23}-\frac{4055816752}{396653711}a^{22}-\frac{1398735}{12795281}a^{21}+\frac{12324590215}{396653711}a^{20}+\frac{102946896}{396653711}a^{19}-\frac{32529118520}{396653711}a^{18}-\frac{827398377}{1586614844}a^{17}+\frac{307098568663}{1586614844}a^{16}+\frac{256994244}{396653711}a^{15}-\frac{152707923952}{396653711}a^{14}+\frac{521935826}{396653711}a^{13}+\frac{181958896466}{396653711}a^{12}+\frac{292801860}{396653711}a^{11}-\frac{203891964320}{396653711}a^{10}-\frac{436312071}{793307422}a^{9}+\frac{201854879888}{396653711}a^{8}+\frac{33196644}{396653711}a^{7}-\frac{143137871736}{396653711}a^{6}-\frac{5035446}{396653711}a^{5}+\frac{4666024476}{396653711}a^{4}+\frac{2606246504}{396653711}a^{3}-\frac{697380256}{396653711}a^{2}-\frac{93249}{396653711}a-\frac{304893151}{396653711}$, $\frac{13075084}{396653711}a^{31}+\frac{16722861}{793307422}a^{30}-\frac{205932573}{793307422}a^{29}-\frac{265267503}{1586614844}a^{28}+\frac{4497850247}{3173229688}a^{27}+\frac{363316308}{396653711}a^{26}-\frac{2647704510}{396653711}a^{25}-\frac{1713943390}{396653711}a^{24}+\frac{11558374256}{396653711}a^{23}+\frac{7489613027}{396653711}a^{22}-\frac{35122944395}{396653711}a^{21}-\frac{22890117215}{396653711}a^{20}+\frac{92702345560}{396653711}a^{19}+\frac{60604310186}{396653711}a^{18}-\frac{875183032653}{1586614844}a^{17}-\frac{573712095703}{1586614844}a^{16}+\frac{435191095856}{396653711}a^{15}+\frac{286633525482}{396653711}a^{14}-\frac{518551293898}{396653711}a^{13}-\frac{348350853106}{396653711}a^{12}+\frac{581056732960}{396653711}a^{11}+\frac{392380966000}{396653711}a^{10}-\frac{575251395664}{396653711}a^{9}-\frac{782555751623}{793307422}a^{8}+\frac{407685694490}{396653711}a^{7}+\frac{284383874116}{396653711}a^{6}-\frac{13297360428}{396653711}a^{5}-\frac{26159325116}{396653711}a^{4}+\frac{1987412768}{396653711}a^{3}+\frac{3957600016}{396653711}a^{2}-\frac{261501680}{396653711}a-\frac{574756080}{396653711}$, $\frac{555795}{793307422}a^{31}+\frac{4588028}{396653711}a^{30}-\frac{6113745}{1586614844}a^{29}-\frac{72261441}{793307422}a^{28}+\frac{7225335}{396653711}a^{27}+\frac{197286598}{396653711}a^{26}-\frac{252886725}{3173229688}a^{25}-\frac{929075670}{396653711}a^{24}+\frac{529098741}{1586614844}a^{23}+\frac{4055816752}{396653711}a^{22}-\frac{8336925}{12795281}a^{21}-\frac{12324590215}{396653711}a^{20}+\frac{613597680}{396653711}a^{19}+\frac{32529118520}{396653711}a^{18}-\frac{4931569035}{1586614844}a^{17}-\frac{307098568663}{1586614844}a^{16}+\frac{1531771020}{396653711}a^{15}+\frac{152707923952}{396653711}a^{14}+\frac{6197252365}{793307422}a^{13}-\frac{181958896466}{396653711}a^{12}+\frac{1745196300}{396653711}a^{11}+\frac{203891964320}{396653711}a^{10}-\frac{2600564805}{793307422}a^{9}-\frac{201854879888}{396653711}a^{8}+\frac{197863020}{396653711}a^{7}+\frac{143137871736}{396653711}a^{6}-\frac{30012930}{396653711}a^{5}-\frac{4666024476}{396653711}a^{4}+\frac{13103243014}{396653711}a^{3}+\frac{697380256}{396653711}a^{2}-\frac{555795}{396653711}a+\frac{304893151}{396653711}$, $\frac{19588389}{396653711}a^{31}+\frac{16722861}{793307422}a^{30}-\frac{1234068507}{3173229688}a^{29}-\frac{265267503}{1586614844}a^{28}+\frac{6738411139}{3173229688}a^{27}+\frac{363316308}{396653711}a^{26}-\frac{7933297545}{793307422}a^{25}-\frac{1713943390}{396653711}a^{24}+\frac{17316135876}{396653711}a^{23}+\frac{7489613027}{396653711}a^{22}-\frac{210477239805}{1586614844}a^{21}-\frac{22890117215}{396653711}a^{20}+\frac{138881678010}{396653711}a^{19}+\frac{60604310186}{396653711}a^{18}-\frac{1311165885063}{1586614844}a^{17}-\frac{573712095703}{1586614844}a^{16}+\frac{651979939476}{396653711}a^{15}+\frac{286633525482}{396653711}a^{14}-\frac{1553731427091}{793307422}a^{13}-\frac{348350853106}{396653711}a^{12}+\frac{870508007160}{396653711}a^{11}+\frac{392380966000}{396653711}a^{10}-\frac{861810762444}{396653711}a^{9}-\frac{782555751623}{793307422}a^{8}+\frac{610279650906}{396653711}a^{7}+\frac{284383874116}{396653711}a^{6}-\frac{19921391613}{396653711}a^{5}-\frac{26159325116}{396653711}a^{4}+\frac{2977435128}{396653711}a^{3}+\frac{3957600016}{396653711}a^{2}-\frac{391767780}{396653711}a-\frac{574756080}{396653711}$, $\frac{279747}{793307422}a^{31}+\frac{4588028}{396653711}a^{30}-\frac{3077217}{1586614844}a^{29}-\frac{72261441}{793307422}a^{28}+\frac{3636711}{396653711}a^{27}+\frac{197286598}{396653711}a^{26}-\frac{127284885}{3173229688}a^{25}-\frac{929075670}{396653711}a^{24}+\frac{66583569}{396653711}a^{23}+\frac{4055816752}{396653711}a^{22}-\frac{4196205}{12795281}a^{21}-\frac{12324590215}{396653711}a^{20}+\frac{308840688}{396653711}a^{19}+\frac{32529118520}{396653711}a^{18}-\frac{2482195131}{1586614844}a^{17}-\frac{307098568663}{1586614844}a^{16}+\frac{770982732}{396653711}a^{15}+\frac{152707923952}{396653711}a^{14}+\frac{1565807478}{396653711}a^{13}-\frac{181958896466}{396653711}a^{12}+\frac{878405580}{396653711}a^{11}+\frac{203891964320}{396653711}a^{10}-\frac{1308936213}{793307422}a^{9}-\frac{201854879888}{396653711}a^{8}+\frac{99589932}{396653711}a^{7}+\frac{143137871736}{396653711}a^{6}-\frac{15106338}{396653711}a^{5}-\frac{4666024476}{396653711}a^{4}+\frac{7422085801}{396653711}a^{3}+\frac{697380256}{396653711}a^{2}-\frac{279747}{396653711}a+\frac{304893151}{396653711}$, $\frac{8836375}{793307422}a^{31}+\frac{5449278}{396653711}a^{30}-\frac{31810950}{396653711}a^{29}-\frac{84663441}{793307422}a^{28}+\frac{666262675}{1586614844}a^{27}+\frac{229755723}{396653711}a^{26}-\frac{6136004913}{3173229688}a^{25}-\frac{8628697255}{3173229688}a^{24}+\frac{3304804250}{396653711}a^{23}+\frac{4700031752}{396653711}a^{22}-\frac{9246382800}{396653711}a^{21}-\frac{14127014215}{396653711}a^{20}+\frac{46753260125}{793307422}a^{19}+\frac{37085992270}{396653711}a^{18}-\frac{52982904500}{396653711}a^{17}-\frac{348411008663}{1586614844}a^{16}+\frac{195319857565}{793307422}a^{15}+\frac{171747240618}{396653711}a^{14}-\frac{77180433800}{396653711}a^{13}-\frac{197003900466}{396653711}a^{12}+\frac{80518816275}{396653711}a^{11}+\frac{219587728820}{396653711}a^{10}-\frac{64894338000}{396653711}a^{9}-\frac{214504919888}{396653711}a^{8}+\frac{9875532700}{396653711}a^{7}+\frac{145062937736}{396653711}a^{6}+\frac{84541182195}{396653711}a^{5}+\frac{12156216242}{396653711}a^{4}+\frac{222676650}{396653711}a^{3}+\frac{740787256}{396653711}a^{2}-\frac{28276400}{396653711}a-\frac{97272560}{396653711}$, $\frac{87690611}{3173229688}a^{31}+\frac{25801661}{1586614844}a^{30}-\frac{684674343}{3173229688}a^{29}-\frac{406227587}{3173229688}a^{28}+\frac{3725870875}{3173229688}a^{27}+\frac{1109270355}{1586614844}a^{26}-\frac{8758022645}{1586614844}a^{25}-\frac{5224533039}{1586614844}a^{24}+\frac{9548713343}{396653711}a^{23}+\frac{5702318850}{396653711}a^{22}-\frac{1860077707}{25590562}a^{21}-\frac{69317588335}{1586614844}a^{20}+\frac{75899112596}{396653711}a^{19}+\frac{183055132725}{1586614844}a^{18}-\frac{714724218657}{1586614844}a^{17}-\frac{108047941812}{396653711}a^{16}+\frac{353729116730}{396653711}a^{15}+\frac{430039350745}{793307422}a^{14}-\frac{826031947093}{793307422}a^{13}-\frac{256706714760}{396653711}a^{12}+\frac{926567529425}{793307422}a^{11}+\frac{577834554315}{793307422}a^{10}-\frac{909867249231}{793307422}a^{9}-\frac{573008793683}{793307422}a^{8}+\frac{314259397255}{396653711}a^{7}+\frac{204758824764}{396653711}a^{6}+\frac{6598536737}{396653711}a^{5}-\frac{10638385777}{396653711}a^{4}+\frac{4176927464}{396653711}a^{3}+\frac{4724967978}{396653711}a^{2}+\frac{658039167}{396653711}a-\frac{130280225}{396653711}$, $\frac{861250}{396653711}a^{31}+\frac{5449278}{396653711}a^{30}-\frac{6201000}{396653711}a^{29}-\frac{84663441}{793307422}a^{28}+\frac{32469125}{396653711}a^{27}+\frac{229755723}{396653711}a^{26}-\frac{1196091895}{3173229688}a^{25}-\frac{8628697255}{3173229688}a^{24}+\frac{644215000}{396653711}a^{23}+\frac{4700031752}{396653711}a^{22}-\frac{1802424000}{396653711}a^{21}-\frac{14127014215}{396653711}a^{20}+\frac{4556873750}{396653711}a^{19}+\frac{37085992270}{396653711}a^{18}-\frac{10328110000}{396653711}a^{17}-\frac{348411008663}{1586614844}a^{16}+\frac{19039316666}{396653711}a^{15}+\frac{171747240618}{396653711}a^{14}-\frac{15045004000}{396653711}a^{13}-\frac{197003900466}{396653711}a^{12}+\frac{15695764500}{396653711}a^{11}+\frac{219587728820}{396653711}a^{10}-\frac{12650040000}{396653711}a^{9}-\frac{214504919888}{396653711}a^{8}+\frac{1925066000}{396653711}a^{7}+\frac{145062937736}{396653711}a^{6}+\frac{16822240718}{396653711}a^{5}+\frac{12156216242}{396653711}a^{4}+\frac{43407000}{396653711}a^{3}+\frac{740787256}{396653711}a^{2}-\frac{5512000}{396653711}a-\frac{97272560}{396653711}$ Copy content Toggle raw display (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:  \( 2660439411454.7925 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 2660439411454.7925 \cdot 85}{10\cdot\sqrt{1267650600228229401496703205376000000000000000000000000}}\cr\approx \mathstrut & 0.118508747741953 \end{aligned}\] (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 + 44*x^28 - 208*x^26 + 910*x^24 - 2800*x^22 + 7440*x^20 - 17664*x^18 + 35492*x^16 - 44096*x^14 + 49952*x^12 - 50240*x^10 + 37432*x^8 - 5696*x^6 + 864*x^4 - 128*x^2 + 16)
 
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 + 44*x^28 - 208*x^26 + 910*x^24 - 2800*x^22 + 7440*x^20 - 17664*x^18 + 35492*x^16 - 44096*x^14 + 49952*x^12 - 50240*x^10 + 37432*x^8 - 5696*x^6 + 864*x^4 - 128*x^2 + 16, 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 + 44*x^28 - 208*x^26 + 910*x^24 - 2800*x^22 + 7440*x^20 - 17664*x^18 + 35492*x^16 - 44096*x^14 + 49952*x^12 - 50240*x^10 + 37432*x^8 - 5696*x^6 + 864*x^4 - 128*x^2 + 16);
 
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 + 44*x^28 - 208*x^26 + 910*x^24 - 2800*x^22 + 7440*x^20 - 17664*x^18 + 35492*x^16 - 44096*x^14 + 49952*x^12 - 50240*x^10 + 37432*x^8 - 5696*x^6 + 864*x^4 - 128*x^2 + 16);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_4\times C_8$ (as 32T43):

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 32
The 32 conjugacy class representatives for $C_4\times C_8$
Character table for $C_4\times C_8$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.0.8000.2, 4.4.51200.1, \(\Q(\zeta_{16})^+\), 4.0.256000.2, 4.0.256000.4, 8.0.64000000.2, 8.8.2621440000.1, 8.0.65536000000.1, 8.8.33554432000000.2, 8.8.33554432000000.1, 8.0.2147483648.1, 8.0.1342177280000.1, 16.0.4294967296000000000000.1, 16.16.1125899906842624000000000000.1, 16.0.1801439850948198400000000.1

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 ${\href{/padicField/3.8.0.1}{8} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.8.0.1}{8} }^{4}$ ${\href{/padicField/13.8.0.1}{8} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.8.0.1}{8} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.8.0.1}{8} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{8}$ ${\href{/padicField/43.8.0.1}{8} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{4}$ ${\href{/padicField/59.8.0.1}{8} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display Deg $32$$8$$4$$124$
\(5\) Copy content Toggle raw display Deg $32$$4$$8$$24$