Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 36*x^30 + 576*x^28 - 5474*x^26 + 34020*x^24 - 130166*x^22 + 187427*x^20 + 377425*x^18 + 1235554*x^16 - 19352440*x^14 + 35136537*x^12 + 25383639*x^10 + 8167525*x^8 + 1500436*x^6 + 160416*x^4 + 8704*x^2 + 256)
gp: K = bnfinit(y^32 - 36*y^30 + 576*y^28 - 5474*y^26 + 34020*y^24 - 130166*y^22 + 187427*y^20 + 377425*y^18 + 1235554*y^16 - 19352440*y^14 + 35136537*y^12 + 25383639*y^10 + 8167525*y^8 + 1500436*y^6 + 160416*y^4 + 8704*y^2 + 256, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 36*x^30 + 576*x^28 - 5474*x^26 + 34020*x^24 - 130166*x^22 + 187427*x^20 + 377425*x^18 + 1235554*x^16 - 19352440*x^14 + 35136537*x^12 + 25383639*x^10 + 8167525*x^8 + 1500436*x^6 + 160416*x^4 + 8704*x^2 + 256);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^32 - 36*x^30 + 576*x^28 - 5474*x^26 + 34020*x^24 - 130166*x^22 + 187427*x^20 + 377425*x^18 + 1235554*x^16 - 19352440*x^14 + 35136537*x^12 + 25383639*x^10 + 8167525*x^8 + 1500436*x^6 + 160416*x^4 + 8704*x^2 + 256)
\( x^{32} - 36 x^{30} + 576 x^{28} - 5474 x^{26} + 34020 x^{24} - 130166 x^{22} + 187427 x^{20} + \cdots + 256 \)
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: |
\(802755791064235699686694954262517410037760000000000000000\)
\(\medspace = 2^{32}\cdot 5^{16}\cdot 7^{16}\cdot 29^{8}\cdot 521^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(60.02\) | 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: | $2\cdot 5^{1/2}7^{1/2}29^{1/2}521^{1/2}\approx 1454.3933443192045$ | ||
| Ramified primes: |
\(2\), \(5\), \(7\), \(29\), \(521\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{20}a^{20}+\frac{1}{20}a^{18}-\frac{1}{2}a^{16}-\frac{2}{5}a^{14}+\frac{1}{4}a^{12}-\frac{3}{20}a^{10}+\frac{1}{4}a^{8}+\frac{1}{10}a^{6}+\frac{1}{20}a^{2}-\frac{1}{5}$, $\frac{1}{20}a^{21}+\frac{1}{20}a^{19}-\frac{1}{2}a^{17}-\frac{2}{5}a^{15}+\frac{1}{4}a^{13}-\frac{3}{20}a^{11}+\frac{1}{4}a^{9}+\frac{1}{10}a^{7}+\frac{1}{20}a^{3}-\frac{1}{5}a$, $\frac{1}{20}a^{22}+\frac{9}{20}a^{18}+\frac{1}{10}a^{16}-\frac{7}{20}a^{14}-\frac{2}{5}a^{12}+\frac{2}{5}a^{10}-\frac{3}{20}a^{8}-\frac{1}{10}a^{6}+\frac{1}{20}a^{4}-\frac{1}{4}a^{2}+\frac{1}{5}$, $\frac{1}{40}a^{23}+\frac{9}{40}a^{19}+\frac{1}{20}a^{17}+\frac{13}{40}a^{15}-\frac{1}{5}a^{13}+\frac{1}{5}a^{11}+\frac{17}{40}a^{9}-\frac{1}{20}a^{7}+\frac{1}{40}a^{5}-\frac{1}{8}a^{3}+\frac{1}{10}a$, $\frac{1}{400}a^{24}-\frac{3}{200}a^{22}-\frac{1}{400}a^{20}+\frac{49}{200}a^{18}+\frac{21}{400}a^{16}+\frac{97}{200}a^{14}+\frac{3}{200}a^{12}-\frac{1}{400}a^{10}-\frac{97}{200}a^{8}-\frac{127}{400}a^{6}-\frac{51}{400}a^{4}+\frac{23}{50}a^{2}+\frac{11}{25}$, $\frac{1}{400}a^{25}+\frac{1}{100}a^{23}-\frac{1}{400}a^{21}-\frac{3}{100}a^{19}-\frac{159}{400}a^{17}-\frac{19}{100}a^{15}-\frac{37}{200}a^{13}-\frac{121}{400}a^{11}+\frac{11}{25}a^{9}+\frac{53}{400}a^{7}-\frac{41}{400}a^{5}+\frac{67}{200}a^{3}+\frac{1}{25}a$, $\frac{1}{400}a^{26}+\frac{3}{400}a^{22}-\frac{1}{50}a^{20}+\frac{69}{400}a^{18}-\frac{1}{2}a^{16}+\frac{9}{40}a^{14}+\frac{3}{80}a^{12}+\frac{1}{20}a^{10}+\frac{89}{400}a^{8}+\frac{107}{400}a^{6}-\frac{41}{200}a^{4}+\frac{9}{20}a^{2}+\frac{1}{25}$, $\frac{1}{800}a^{27}-\frac{1}{800}a^{25}+\frac{9}{800}a^{23}+\frac{13}{800}a^{21}-\frac{9}{800}a^{19}-\frac{21}{800}a^{17}-\frac{33}{100}a^{15}-\frac{291}{800}a^{13}-\frac{39}{800}a^{11}-\frac{17}{800}a^{9}+\frac{137}{400}a^{7}-\frac{31}{800}a^{5}-\frac{12}{25}a^{3}+\frac{1}{5}a$, $\frac{1}{123200}a^{28}+\frac{127}{123200}a^{26}+\frac{101}{123200}a^{24}+\frac{1}{24640}a^{22}+\frac{9}{3520}a^{20}+\frac{4937}{11200}a^{18}+\frac{897}{30800}a^{16}-\frac{1633}{11200}a^{14}-\frac{46047}{123200}a^{12}-\frac{18189}{123200}a^{10}-\frac{11331}{61600}a^{8}+\frac{43741}{123200}a^{6}+\frac{3747}{30800}a^{4}-\frac{213}{1925}a^{2}-\frac{184}{385}$, $\frac{1}{123200}a^{29}-\frac{27}{123200}a^{27}-\frac{53}{123200}a^{25}+\frac{467}{123200}a^{23}-\frac{197}{17600}a^{21}+\frac{2319}{11200}a^{19}+\frac{177}{61600}a^{17}-\frac{3369}{11200}a^{15}-\frac{3081}{123200}a^{13}-\frac{475}{4928}a^{11}+\frac{9927}{30800}a^{9}+\frac{40661}{123200}a^{7}+\frac{3547}{12320}a^{5}-\frac{349}{3850}a^{3}-\frac{227}{1925}a$, $\frac{1}{54\!\cdots\!00}a^{30}+\frac{29\!\cdots\!51}{78\!\cdots\!00}a^{28}+\frac{58\!\cdots\!91}{54\!\cdots\!00}a^{26}-\frac{33\!\cdots\!81}{54\!\cdots\!00}a^{24}-\frac{12\!\cdots\!99}{54\!\cdots\!00}a^{22}+\frac{12\!\cdots\!53}{54\!\cdots\!00}a^{20}-\frac{12\!\cdots\!53}{54\!\cdots\!60}a^{18}+\frac{23\!\cdots\!41}{54\!\cdots\!00}a^{16}+\frac{13\!\cdots\!51}{11\!\cdots\!00}a^{14}-\frac{43\!\cdots\!79}{10\!\cdots\!20}a^{12}+\frac{16\!\cdots\!63}{12\!\cdots\!75}a^{10}+\frac{17\!\cdots\!41}{10\!\cdots\!20}a^{8}+\frac{18\!\cdots\!27}{99\!\cdots\!52}a^{6}+\frac{11\!\cdots\!83}{68\!\cdots\!00}a^{4}-\frac{10\!\cdots\!27}{44\!\cdots\!00}a^{2}-\frac{21\!\cdots\!19}{85\!\cdots\!25}$, $\frac{1}{10\!\cdots\!00}a^{31}-\frac{42\!\cdots\!61}{19\!\cdots\!00}a^{29}+\frac{33\!\cdots\!48}{17\!\cdots\!05}a^{27}+\frac{58\!\cdots\!43}{10\!\cdots\!20}a^{25}+\frac{17\!\cdots\!03}{27\!\cdots\!00}a^{23}-\frac{87\!\cdots\!03}{54\!\cdots\!00}a^{21}-\frac{13\!\cdots\!69}{10\!\cdots\!00}a^{19}+\frac{31\!\cdots\!41}{10\!\cdots\!00}a^{17}-\frac{77\!\cdots\!59}{78\!\cdots\!00}a^{15}+\frac{95\!\cdots\!53}{27\!\cdots\!68}a^{13}-\frac{13\!\cdots\!69}{15\!\cdots\!00}a^{11}+\frac{49\!\cdots\!59}{21\!\cdots\!40}a^{9}+\frac{14\!\cdots\!89}{90\!\cdots\!00}a^{7}+\frac{34\!\cdots\!37}{85\!\cdots\!25}a^{5}+\frac{19\!\cdots\!27}{12\!\cdots\!00}a^{3}-\frac{39\!\cdots\!34}{85\!\cdots\!25}a$
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
$C_{2}\times C_{2}\times C_{84}$, which has order $336$ (assuming GRH)
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Relative class number: $336$ (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: |
\( \frac{5945630903775229538143}{4242200574267219677565760} a^{31} - \frac{3845098749743501047643}{75753581683343208527960} a^{29} + \frac{8677572088192427724277353}{10605501435668049193914400} a^{27} - \frac{5201750661132708003938721}{662843839729253074619650} a^{25} + \frac{104693730651633492521389131}{2121100287133609838782880} a^{23} - \frac{255770620815320018774002251}{1325687679458506149239300} a^{21} + \frac{6435020180840887438115772143}{21211002871336098387828800} a^{19} + \frac{9908962750212619757195042613}{21211002871336098387828800} a^{17} + \frac{2462300047752335071601892423}{1515071633666864170559200} a^{15} - \frac{291469780144098682827657848691}{10605501435668049193914400} a^{13} + \frac{167094083982982171509378618891}{3030143267333728341118400} a^{11} + \frac{513284839738472348017290010487}{21211002871336098387828800} a^{9} + \frac{10010186315825421500695109901}{1928272988303281671620800} a^{7} + \frac{215458182764132067667005041}{424220057426721967756576} a^{5} - \frac{266796412822408368901553}{34433446219701458421800} a^{3} - \frac{2827473926570193786666923}{662843839729253074619650} a \)
(order $4$)
| 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: |
$\frac{44\!\cdots\!07}{73\!\cdots\!00}a^{30}-\frac{22\!\cdots\!27}{10\!\cdots\!20}a^{28}+\frac{12\!\cdots\!99}{36\!\cdots\!00}a^{26}-\frac{28\!\cdots\!47}{91\!\cdots\!00}a^{24}+\frac{67\!\cdots\!67}{36\!\cdots\!00}a^{22}-\frac{29\!\cdots\!51}{45\!\cdots\!00}a^{20}+\frac{41\!\cdots\!03}{73\!\cdots\!00}a^{18}+\frac{47\!\cdots\!31}{14\!\cdots\!80}a^{16}+\frac{23\!\cdots\!93}{26\!\cdots\!80}a^{14}-\frac{41\!\cdots\!11}{36\!\cdots\!00}a^{12}+\frac{19\!\cdots\!01}{14\!\cdots\!00}a^{10}+\frac{23\!\cdots\!49}{73\!\cdots\!00}a^{8}+\frac{86\!\cdots\!49}{73\!\cdots\!00}a^{6}+\frac{15\!\cdots\!81}{91\!\cdots\!00}a^{4}+\frac{13\!\cdots\!63}{13\!\cdots\!40}a^{2}-\frac{14\!\cdots\!22}{11\!\cdots\!75}$, $\frac{65\!\cdots\!37}{10\!\cdots\!00}a^{31}-\frac{16\!\cdots\!03}{78\!\cdots\!00}a^{29}+\frac{19\!\cdots\!17}{54\!\cdots\!00}a^{27}-\frac{91\!\cdots\!51}{27\!\cdots\!00}a^{25}+\frac{45\!\cdots\!45}{21\!\cdots\!44}a^{23}-\frac{44\!\cdots\!59}{54\!\cdots\!60}a^{21}+\frac{13\!\cdots\!01}{10\!\cdots\!00}a^{19}+\frac{22\!\cdots\!53}{10\!\cdots\!00}a^{17}+\frac{27\!\cdots\!29}{39\!\cdots\!00}a^{15}-\frac{64\!\cdots\!97}{54\!\cdots\!00}a^{13}+\frac{36\!\cdots\!29}{15\!\cdots\!00}a^{11}+\frac{11\!\cdots\!27}{10\!\cdots\!00}a^{9}+\frac{31\!\cdots\!29}{99\!\cdots\!00}a^{7}+\frac{39\!\cdots\!83}{68\!\cdots\!00}a^{5}+\frac{11\!\cdots\!61}{17\!\cdots\!92}a^{3}+\frac{87\!\cdots\!29}{17\!\cdots\!50}a$, $\frac{23\!\cdots\!07}{17\!\cdots\!00}a^{30}-\frac{15\!\cdots\!57}{31\!\cdots\!00}a^{28}+\frac{34\!\cdots\!87}{44\!\cdots\!00}a^{26}-\frac{65\!\cdots\!57}{88\!\cdots\!00}a^{24}+\frac{25\!\cdots\!11}{55\!\cdots\!50}a^{22}-\frac{15\!\cdots\!31}{88\!\cdots\!00}a^{20}+\frac{47\!\cdots\!17}{17\!\cdots\!00}a^{18}+\frac{85\!\cdots\!23}{17\!\cdots\!00}a^{16}+\frac{20\!\cdots\!43}{12\!\cdots\!00}a^{14}-\frac{11\!\cdots\!21}{44\!\cdots\!00}a^{12}+\frac{17\!\cdots\!47}{36\!\cdots\!00}a^{10}+\frac{52\!\cdots\!13}{17\!\cdots\!00}a^{8}+\frac{13\!\cdots\!27}{17\!\cdots\!00}a^{6}+\frac{97\!\cdots\!09}{88\!\cdots\!20}a^{4}+\frac{53\!\cdots\!43}{78\!\cdots\!35}a^{2}-\frac{84\!\cdots\!22}{55\!\cdots\!45}$, $\frac{13\!\cdots\!77}{24\!\cdots\!00}a^{31}-\frac{90\!\cdots\!21}{15\!\cdots\!00}a^{30}-\frac{13\!\cdots\!57}{70\!\cdots\!00}a^{29}+\frac{11\!\cdots\!19}{54\!\cdots\!00}a^{28}+\frac{14\!\cdots\!81}{49\!\cdots\!00}a^{27}-\frac{32\!\cdots\!09}{94\!\cdots\!00}a^{26}-\frac{13\!\cdots\!09}{49\!\cdots\!00}a^{25}+\frac{24\!\cdots\!43}{75\!\cdots\!00}a^{24}+\frac{16\!\cdots\!03}{99\!\cdots\!20}a^{23}-\frac{15\!\cdots\!59}{75\!\cdots\!80}a^{22}-\frac{29\!\cdots\!03}{49\!\cdots\!00}a^{21}+\frac{58\!\cdots\!17}{75\!\cdots\!00}a^{20}+\frac{30\!\cdots\!71}{49\!\cdots\!00}a^{19}-\frac{16\!\cdots\!23}{15\!\cdots\!00}a^{18}+\frac{66\!\cdots\!37}{24\!\cdots\!80}a^{17}-\frac{34\!\cdots\!73}{15\!\cdots\!00}a^{16}+\frac{10\!\cdots\!99}{14\!\cdots\!60}a^{15}-\frac{79\!\cdots\!83}{10\!\cdots\!00}a^{14}-\frac{50\!\cdots\!49}{49\!\cdots\!00}a^{13}+\frac{27\!\cdots\!17}{23\!\cdots\!25}a^{12}+\frac{93\!\cdots\!01}{70\!\cdots\!00}a^{11}-\frac{64\!\cdots\!09}{30\!\cdots\!00}a^{10}+\frac{12\!\cdots\!81}{49\!\cdots\!60}a^{9}-\frac{23\!\cdots\!31}{15\!\cdots\!00}a^{8}+\frac{49\!\cdots\!11}{49\!\cdots\!00}a^{7}-\frac{13\!\cdots\!43}{27\!\cdots\!20}a^{6}+\frac{55\!\cdots\!63}{24\!\cdots\!00}a^{5}-\frac{74\!\cdots\!41}{94\!\cdots\!00}a^{4}+\frac{23\!\cdots\!61}{88\!\cdots\!00}a^{3}-\frac{18\!\cdots\!97}{24\!\cdots\!60}a^{2}+\frac{92\!\cdots\!51}{77\!\cdots\!75}a-\frac{43\!\cdots\!16}{23\!\cdots\!25}$, $\frac{33\!\cdots\!57}{10\!\cdots\!00}a^{31}+\frac{13\!\cdots\!49}{75\!\cdots\!00}a^{30}-\frac{84\!\cdots\!73}{78\!\cdots\!00}a^{29}-\frac{35\!\cdots\!65}{54\!\cdots\!92}a^{28}+\frac{95\!\cdots\!71}{54\!\cdots\!00}a^{27}+\frac{19\!\cdots\!59}{18\!\cdots\!00}a^{26}-\frac{56\!\cdots\!97}{34\!\cdots\!10}a^{25}-\frac{23\!\cdots\!11}{23\!\cdots\!25}a^{24}+\frac{11\!\cdots\!59}{10\!\cdots\!20}a^{23}+\frac{11\!\cdots\!57}{18\!\cdots\!00}a^{22}-\frac{53\!\cdots\!61}{13\!\cdots\!00}a^{21}-\frac{57\!\cdots\!61}{23\!\cdots\!25}a^{20}+\frac{12\!\cdots\!97}{21\!\cdots\!40}a^{19}+\frac{27\!\cdots\!51}{75\!\cdots\!00}a^{18}+\frac{12\!\cdots\!17}{10\!\cdots\!00}a^{17}+\frac{97\!\cdots\!23}{15\!\cdots\!60}a^{16}+\frac{18\!\cdots\!77}{48\!\cdots\!00}a^{15}+\frac{23\!\cdots\!83}{10\!\cdots\!40}a^{14}-\frac{63\!\cdots\!79}{10\!\cdots\!20}a^{13}-\frac{33\!\cdots\!53}{94\!\cdots\!00}a^{12}+\frac{16\!\cdots\!17}{15\!\cdots\!00}a^{11}+\frac{10\!\cdots\!07}{15\!\cdots\!00}a^{10}+\frac{84\!\cdots\!03}{10\!\cdots\!00}a^{9}+\frac{29\!\cdots\!83}{75\!\cdots\!00}a^{8}+\frac{23\!\cdots\!17}{99\!\cdots\!00}a^{7}+\frac{71\!\cdots\!93}{68\!\cdots\!00}a^{6}+\frac{57\!\cdots\!51}{13\!\cdots\!00}a^{5}+\frac{58\!\cdots\!33}{37\!\cdots\!00}a^{4}+\frac{45\!\cdots\!41}{11\!\cdots\!25}a^{3}+\frac{29\!\cdots\!51}{24\!\cdots\!60}a^{2}+\frac{32\!\cdots\!17}{17\!\cdots\!50}a+\frac{66\!\cdots\!42}{23\!\cdots\!25}$, $\frac{15\!\cdots\!27}{10\!\cdots\!00}a^{31}-\frac{48\!\cdots\!71}{10\!\cdots\!00}a^{30}-\frac{38\!\cdots\!33}{78\!\cdots\!00}a^{29}+\frac{50\!\cdots\!07}{30\!\cdots\!00}a^{28}+\frac{43\!\cdots\!73}{54\!\cdots\!00}a^{27}-\frac{56\!\cdots\!33}{21\!\cdots\!00}a^{26}-\frac{10\!\cdots\!21}{13\!\cdots\!00}a^{25}+\frac{54\!\cdots\!97}{21\!\cdots\!00}a^{24}+\frac{51\!\cdots\!33}{10\!\cdots\!20}a^{23}-\frac{68\!\cdots\!03}{43\!\cdots\!60}a^{22}-\frac{15\!\cdots\!49}{85\!\cdots\!25}a^{21}+\frac{13\!\cdots\!03}{21\!\cdots\!00}a^{20}+\frac{28\!\cdots\!27}{10\!\cdots\!00}a^{19}-\frac{21\!\cdots\!91}{21\!\cdots\!00}a^{18}+\frac{56\!\cdots\!99}{10\!\cdots\!00}a^{17}-\frac{16\!\cdots\!33}{10\!\cdots\!00}a^{16}+\frac{66\!\cdots\!47}{39\!\cdots\!00}a^{15}-\frac{15\!\cdots\!17}{30\!\cdots\!00}a^{14}-\frac{14\!\cdots\!49}{54\!\cdots\!00}a^{13}+\frac{19\!\cdots\!81}{21\!\cdots\!00}a^{12}+\frac{76\!\cdots\!71}{15\!\cdots\!00}a^{11}-\frac{54\!\cdots\!41}{30\!\cdots\!00}a^{10}+\frac{37\!\cdots\!01}{10\!\cdots\!00}a^{9}-\frac{41\!\cdots\!73}{54\!\cdots\!00}a^{8}+\frac{10\!\cdots\!31}{99\!\cdots\!00}a^{7}-\frac{41\!\cdots\!51}{39\!\cdots\!60}a^{6}+\frac{47\!\cdots\!97}{27\!\cdots\!00}a^{5}-\frac{14\!\cdots\!03}{54\!\cdots\!00}a^{4}+\frac{10\!\cdots\!57}{88\!\cdots\!00}a^{3}+\frac{55\!\cdots\!63}{35\!\cdots\!80}a^{2}-\frac{12\!\cdots\!82}{85\!\cdots\!25}a+\frac{68\!\cdots\!18}{33\!\cdots\!75}$, $\frac{80\!\cdots\!63}{15\!\cdots\!60}a^{31}+\frac{19\!\cdots\!27}{25\!\cdots\!84}a^{30}-\frac{10\!\cdots\!13}{55\!\cdots\!20}a^{29}-\frac{12\!\cdots\!91}{45\!\cdots\!00}a^{28}+\frac{23\!\cdots\!17}{78\!\cdots\!80}a^{27}+\frac{14\!\cdots\!27}{31\!\cdots\!00}a^{26}-\frac{22\!\cdots\!65}{78\!\cdots\!48}a^{25}-\frac{34\!\cdots\!11}{79\!\cdots\!00}a^{24}+\frac{13\!\cdots\!31}{78\!\cdots\!80}a^{23}+\frac{17\!\cdots\!03}{63\!\cdots\!60}a^{22}-\frac{26\!\cdots\!31}{39\!\cdots\!40}a^{21}-\frac{41\!\cdots\!47}{39\!\cdots\!00}a^{20}+\frac{15\!\cdots\!27}{15\!\cdots\!60}a^{19}+\frac{10\!\cdots\!23}{63\!\cdots\!00}a^{18}+\frac{29\!\cdots\!29}{15\!\cdots\!60}a^{17}+\frac{17\!\cdots\!39}{63\!\cdots\!00}a^{16}+\frac{69\!\cdots\!93}{11\!\cdots\!40}a^{15}+\frac{10\!\cdots\!21}{11\!\cdots\!00}a^{14}-\frac{77\!\cdots\!87}{78\!\cdots\!80}a^{13}-\frac{48\!\cdots\!03}{31\!\cdots\!00}a^{12}+\frac{41\!\cdots\!91}{22\!\cdots\!80}a^{11}+\frac{53\!\cdots\!79}{18\!\cdots\!60}a^{10}+\frac{36\!\cdots\!71}{31\!\cdots\!92}a^{9}+\frac{96\!\cdots\!77}{63\!\cdots\!00}a^{8}+\frac{45\!\cdots\!49}{14\!\cdots\!60}a^{7}+\frac{24\!\cdots\!03}{58\!\cdots\!00}a^{6}+\frac{38\!\cdots\!93}{78\!\cdots\!80}a^{5}+\frac{53\!\cdots\!81}{79\!\cdots\!00}a^{4}+\frac{41\!\cdots\!49}{11\!\cdots\!80}a^{3}+\frac{35\!\cdots\!41}{51\!\cdots\!00}a^{2}+\frac{32\!\cdots\!68}{48\!\cdots\!03}a+\frac{37\!\cdots\!67}{99\!\cdots\!25}$, $\frac{31\!\cdots\!47}{78\!\cdots\!00}a^{31}+\frac{11\!\cdots\!11}{54\!\cdots\!00}a^{30}-\frac{11\!\cdots\!17}{78\!\cdots\!00}a^{29}-\frac{84\!\cdots\!33}{11\!\cdots\!40}a^{28}+\frac{18\!\cdots\!47}{78\!\cdots\!00}a^{27}+\frac{67\!\cdots\!33}{54\!\cdots\!60}a^{26}-\frac{35\!\cdots\!89}{15\!\cdots\!60}a^{25}-\frac{16\!\cdots\!73}{13\!\cdots\!00}a^{24}+\frac{21\!\cdots\!81}{15\!\cdots\!60}a^{23}+\frac{20\!\cdots\!57}{27\!\cdots\!00}a^{22}-\frac{42\!\cdots\!43}{78\!\cdots\!00}a^{21}-\frac{39\!\cdots\!73}{13\!\cdots\!00}a^{20}+\frac{79\!\cdots\!53}{97\!\cdots\!60}a^{19}+\frac{24\!\cdots\!51}{54\!\cdots\!00}a^{18}+\frac{11\!\cdots\!97}{78\!\cdots\!00}a^{17}+\frac{38\!\cdots\!71}{54\!\cdots\!00}a^{16}+\frac{37\!\cdots\!53}{78\!\cdots\!00}a^{15}+\frac{11\!\cdots\!27}{48\!\cdots\!00}a^{14}-\frac{24\!\cdots\!23}{31\!\cdots\!92}a^{13}-\frac{45\!\cdots\!85}{10\!\cdots\!72}a^{12}+\frac{58\!\cdots\!37}{39\!\cdots\!00}a^{11}+\frac{64\!\cdots\!63}{78\!\cdots\!00}a^{10}+\frac{69\!\cdots\!53}{78\!\cdots\!00}a^{9}+\frac{20\!\cdots\!57}{54\!\cdots\!00}a^{8}+\frac{79\!\cdots\!41}{35\!\cdots\!00}a^{7}+\frac{45\!\cdots\!27}{49\!\cdots\!00}a^{6}+\frac{13\!\cdots\!59}{39\!\cdots\!00}a^{5}+\frac{15\!\cdots\!39}{13\!\cdots\!00}a^{4}+\frac{44\!\cdots\!53}{22\!\cdots\!50}a^{3}+\frac{29\!\cdots\!27}{44\!\cdots\!00}a^{2}+\frac{14\!\cdots\!09}{24\!\cdots\!50}a+\frac{20\!\cdots\!92}{85\!\cdots\!25}$, $\frac{98\!\cdots\!89}{10\!\cdots\!00}a^{31}+\frac{11\!\cdots\!33}{27\!\cdots\!00}a^{30}-\frac{12\!\cdots\!53}{39\!\cdots\!00}a^{29}-\frac{16\!\cdots\!57}{11\!\cdots\!00}a^{28}+\frac{14\!\cdots\!51}{27\!\cdots\!00}a^{27}+\frac{12\!\cdots\!09}{54\!\cdots\!00}a^{26}-\frac{54\!\cdots\!27}{10\!\cdots\!20}a^{25}-\frac{12\!\cdots\!09}{54\!\cdots\!00}a^{24}+\frac{42\!\cdots\!93}{13\!\cdots\!40}a^{23}+\frac{77\!\cdots\!63}{54\!\cdots\!00}a^{22}-\frac{65\!\cdots\!13}{54\!\cdots\!00}a^{21}-\frac{30\!\cdots\!63}{54\!\cdots\!00}a^{20}+\frac{39\!\cdots\!31}{21\!\cdots\!40}a^{19}+\frac{47\!\cdots\!23}{54\!\cdots\!00}a^{18}+\frac{35\!\cdots\!09}{10\!\cdots\!00}a^{17}+\frac{37\!\cdots\!09}{27\!\cdots\!00}a^{16}+\frac{84\!\cdots\!39}{78\!\cdots\!00}a^{15}+\frac{37\!\cdots\!81}{78\!\cdots\!00}a^{14}-\frac{96\!\cdots\!99}{54\!\cdots\!60}a^{13}-\frac{43\!\cdots\!93}{54\!\cdots\!00}a^{12}+\frac{52\!\cdots\!79}{15\!\cdots\!00}a^{11}+\frac{12\!\cdots\!21}{78\!\cdots\!00}a^{10}+\frac{21\!\cdots\!31}{10\!\cdots\!00}a^{9}+\frac{10\!\cdots\!11}{13\!\cdots\!00}a^{8}+\frac{52\!\cdots\!99}{99\!\cdots\!00}a^{7}+\frac{91\!\cdots\!07}{49\!\cdots\!00}a^{6}+\frac{54\!\cdots\!41}{68\!\cdots\!00}a^{5}+\frac{82\!\cdots\!21}{34\!\cdots\!00}a^{4}+\frac{59\!\cdots\!12}{11\!\cdots\!25}a^{3}+\frac{70\!\cdots\!27}{44\!\cdots\!00}a^{2}+\frac{14\!\cdots\!22}{85\!\cdots\!25}a+\frac{46\!\cdots\!21}{85\!\cdots\!25}$, $\frac{26\!\cdots\!11}{54\!\cdots\!00}a^{31}+\frac{74\!\cdots\!57}{27\!\cdots\!00}a^{30}-\frac{13\!\cdots\!03}{78\!\cdots\!00}a^{29}-\frac{38\!\cdots\!29}{39\!\cdots\!00}a^{28}+\frac{14\!\cdots\!81}{54\!\cdots\!00}a^{27}+\frac{41\!\cdots\!83}{27\!\cdots\!00}a^{26}-\frac{28\!\cdots\!87}{10\!\cdots\!20}a^{25}-\frac{39\!\cdots\!87}{27\!\cdots\!00}a^{24}+\frac{17\!\cdots\!87}{10\!\cdots\!20}a^{23}+\frac{47\!\cdots\!89}{54\!\cdots\!60}a^{22}-\frac{32\!\cdots\!69}{54\!\cdots\!00}a^{21}-\frac{85\!\cdots\!81}{27\!\cdots\!00}a^{20}+\frac{40\!\cdots\!49}{54\!\cdots\!60}a^{19}+\frac{46\!\cdots\!91}{13\!\cdots\!00}a^{18}+\frac{11\!\cdots\!31}{54\!\cdots\!00}a^{17}+\frac{36\!\cdots\!63}{27\!\cdots\!00}a^{16}+\frac{48\!\cdots\!67}{78\!\cdots\!00}a^{15}+\frac{14\!\cdots\!81}{39\!\cdots\!00}a^{14}-\frac{10\!\cdots\!81}{10\!\cdots\!20}a^{13}-\frac{14\!\cdots\!99}{27\!\cdots\!00}a^{12}+\frac{18\!\cdots\!29}{12\!\cdots\!75}a^{11}+\frac{14\!\cdots\!11}{19\!\cdots\!00}a^{10}+\frac{90\!\cdots\!39}{54\!\cdots\!00}a^{9}+\frac{33\!\cdots\!81}{27\!\cdots\!00}a^{8}+\frac{13\!\cdots\!93}{24\!\cdots\!00}a^{7}+\frac{20\!\cdots\!01}{62\!\cdots\!00}a^{6}+\frac{57\!\cdots\!13}{68\!\cdots\!00}a^{5}+\frac{16\!\cdots\!63}{27\!\cdots\!80}a^{4}+\frac{36\!\cdots\!79}{44\!\cdots\!00}a^{3}+\frac{58\!\cdots\!76}{11\!\cdots\!25}a^{2}+\frac{10\!\cdots\!97}{17\!\cdots\!50}a+\frac{70\!\cdots\!22}{34\!\cdots\!21}$, $\frac{23\!\cdots\!31}{10\!\cdots\!20}a^{30}-\frac{15\!\cdots\!13}{19\!\cdots\!00}a^{28}+\frac{16\!\cdots\!91}{13\!\cdots\!00}a^{26}-\frac{31\!\cdots\!37}{27\!\cdots\!00}a^{24}+\frac{48\!\cdots\!17}{68\!\cdots\!00}a^{22}-\frac{73\!\cdots\!43}{27\!\cdots\!00}a^{20}+\frac{38\!\cdots\!13}{10\!\cdots\!20}a^{18}+\frac{49\!\cdots\!43}{54\!\cdots\!00}a^{16}+\frac{15\!\cdots\!99}{55\!\cdots\!00}a^{14}-\frac{56\!\cdots\!23}{13\!\cdots\!00}a^{12}+\frac{53\!\cdots\!73}{78\!\cdots\!00}a^{10}+\frac{38\!\cdots\!61}{54\!\cdots\!00}a^{8}+\frac{30\!\cdots\!89}{19\!\cdots\!04}a^{6}+\frac{34\!\cdots\!77}{17\!\cdots\!50}a^{4}+\frac{56\!\cdots\!89}{44\!\cdots\!00}a^{2}+\frac{38\!\cdots\!53}{85\!\cdots\!25}$, $\frac{19\!\cdots\!09}{39\!\cdots\!00}a^{31}+\frac{61\!\cdots\!79}{45\!\cdots\!00}a^{30}-\frac{17\!\cdots\!03}{97\!\cdots\!00}a^{29}-\frac{35\!\cdots\!49}{70\!\cdots\!00}a^{28}+\frac{34\!\cdots\!99}{12\!\cdots\!75}a^{27}+\frac{39\!\cdots\!19}{49\!\cdots\!00}a^{26}-\frac{26\!\cdots\!11}{97\!\cdots\!00}a^{25}-\frac{37\!\cdots\!33}{49\!\cdots\!00}a^{24}+\frac{12\!\cdots\!01}{69\!\cdots\!00}a^{23}+\frac{23\!\cdots\!73}{49\!\cdots\!00}a^{22}-\frac{13\!\cdots\!39}{19\!\cdots\!20}a^{21}-\frac{89\!\cdots\!11}{49\!\cdots\!00}a^{20}+\frac{41\!\cdots\!51}{39\!\cdots\!00}a^{19}+\frac{59\!\cdots\!01}{22\!\cdots\!00}a^{18}+\frac{63\!\cdots\!99}{39\!\cdots\!00}a^{17}+\frac{25\!\cdots\!13}{49\!\cdots\!00}a^{16}+\frac{11\!\cdots\!43}{19\!\cdots\!00}a^{15}+\frac{10\!\cdots\!31}{64\!\cdots\!00}a^{14}-\frac{18\!\cdots\!17}{19\!\cdots\!20}a^{13}-\frac{13\!\cdots\!11}{49\!\cdots\!00}a^{12}+\frac{75\!\cdots\!31}{39\!\cdots\!00}a^{11}+\frac{87\!\cdots\!61}{17\!\cdots\!00}a^{10}+\frac{32\!\cdots\!11}{39\!\cdots\!00}a^{9}+\frac{16\!\cdots\!37}{49\!\cdots\!00}a^{8}+\frac{76\!\cdots\!13}{35\!\cdots\!00}a^{7}+\frac{22\!\cdots\!97}{24\!\cdots\!00}a^{6}+\frac{53\!\cdots\!29}{19\!\cdots\!00}a^{5}+\frac{45\!\cdots\!97}{31\!\cdots\!00}a^{4}+\frac{79\!\cdots\!83}{44\!\cdots\!00}a^{3}+\frac{21\!\cdots\!71}{22\!\cdots\!50}a^{2}+\frac{39\!\cdots\!41}{69\!\cdots\!90}a+\frac{54\!\cdots\!86}{15\!\cdots\!55}$, $\frac{57\!\cdots\!63}{78\!\cdots\!00}a^{31}+\frac{20\!\cdots\!87}{10\!\cdots\!20}a^{30}-\frac{20\!\cdots\!21}{78\!\cdots\!00}a^{29}-\frac{66\!\cdots\!37}{97\!\cdots\!00}a^{28}+\frac{66\!\cdots\!41}{15\!\cdots\!60}a^{27}+\frac{75\!\cdots\!19}{68\!\cdots\!00}a^{26}-\frac{31\!\cdots\!31}{78\!\cdots\!00}a^{25}-\frac{28\!\cdots\!11}{27\!\cdots\!00}a^{24}+\frac{19\!\cdots\!87}{78\!\cdots\!00}a^{23}+\frac{89\!\cdots\!27}{13\!\cdots\!00}a^{22}-\frac{75\!\cdots\!13}{78\!\cdots\!00}a^{21}-\frac{68\!\cdots\!69}{27\!\cdots\!00}a^{20}+\frac{54\!\cdots\!81}{39\!\cdots\!00}a^{19}+\frac{40\!\cdots\!81}{10\!\cdots\!20}a^{18}+\frac{21\!\cdots\!91}{78\!\cdots\!00}a^{17}+\frac{37\!\cdots\!39}{54\!\cdots\!00}a^{16}+\frac{70\!\cdots\!89}{78\!\cdots\!00}a^{15}+\frac{89\!\cdots\!59}{39\!\cdots\!00}a^{14}-\frac{11\!\cdots\!77}{78\!\cdots\!00}a^{13}-\frac{25\!\cdots\!67}{68\!\cdots\!00}a^{12}+\frac{50\!\cdots\!69}{19\!\cdots\!00}a^{11}+\frac{54\!\cdots\!89}{78\!\cdots\!00}a^{10}+\frac{14\!\cdots\!07}{78\!\cdots\!00}a^{9}+\frac{23\!\cdots\!93}{54\!\cdots\!00}a^{8}+\frac{19\!\cdots\!01}{35\!\cdots\!00}a^{7}+\frac{12\!\cdots\!01}{99\!\cdots\!20}a^{6}+\frac{17\!\cdots\!63}{19\!\cdots\!20}a^{5}+\frac{13\!\cdots\!49}{68\!\cdots\!00}a^{4}+\frac{74\!\cdots\!11}{88\!\cdots\!00}a^{3}+\frac{42\!\cdots\!61}{22\!\cdots\!50}a^{2}+\frac{71\!\cdots\!11}{24\!\cdots\!50}a+\frac{53\!\cdots\!49}{85\!\cdots\!25}$, $\frac{12\!\cdots\!47}{10\!\cdots\!00}a^{31}+\frac{10\!\cdots\!73}{15\!\cdots\!96}a^{30}-\frac{31\!\cdots\!89}{78\!\cdots\!00}a^{29}-\frac{29\!\cdots\!34}{12\!\cdots\!75}a^{28}+\frac{34\!\cdots\!31}{54\!\cdots\!00}a^{27}+\frac{77\!\cdots\!19}{19\!\cdots\!20}a^{26}-\frac{16\!\cdots\!19}{27\!\cdots\!00}a^{25}-\frac{52\!\cdots\!83}{13\!\cdots\!00}a^{24}+\frac{20\!\cdots\!63}{54\!\cdots\!00}a^{23}+\frac{23\!\cdots\!71}{97\!\cdots\!00}a^{22}-\frac{15\!\cdots\!03}{10\!\cdots\!72}a^{21}-\frac{90\!\cdots\!93}{97\!\cdots\!00}a^{20}+\frac{23\!\cdots\!03}{10\!\cdots\!00}a^{19}+\frac{15\!\cdots\!49}{11\!\cdots\!40}a^{18}+\frac{45\!\cdots\!23}{10\!\cdots\!00}a^{17}+\frac{18\!\cdots\!51}{78\!\cdots\!80}a^{16}+\frac{52\!\cdots\!29}{39\!\cdots\!00}a^{15}+\frac{31\!\cdots\!49}{39\!\cdots\!40}a^{14}-\frac{11\!\cdots\!67}{54\!\cdots\!00}a^{13}-\frac{12\!\cdots\!69}{97\!\cdots\!60}a^{12}+\frac{61\!\cdots\!31}{15\!\cdots\!00}a^{11}+\frac{10\!\cdots\!03}{39\!\cdots\!00}a^{10}+\frac{29\!\cdots\!81}{10\!\cdots\!00}a^{9}+\frac{50\!\cdots\!79}{39\!\cdots\!00}a^{8}+\frac{76\!\cdots\!23}{99\!\cdots\!00}a^{7}+\frac{11\!\cdots\!61}{35\!\cdots\!00}a^{6}+\frac{88\!\cdots\!37}{68\!\cdots\!00}a^{5}+\frac{25\!\cdots\!97}{55\!\cdots\!20}a^{4}+\frac{47\!\cdots\!19}{44\!\cdots\!00}a^{3}+\frac{70\!\cdots\!57}{22\!\cdots\!50}a^{2}+\frac{83\!\cdots\!91}{17\!\cdots\!50}a+\frac{18\!\cdots\!62}{24\!\cdots\!15}$, $\frac{59\!\cdots\!71}{10\!\cdots\!00}a^{31}+\frac{29\!\cdots\!87}{97\!\cdots\!00}a^{30}-\frac{76\!\cdots\!59}{39\!\cdots\!00}a^{29}-\frac{86\!\cdots\!31}{78\!\cdots\!00}a^{28}+\frac{85\!\cdots\!91}{27\!\cdots\!00}a^{27}+\frac{13\!\cdots\!11}{78\!\cdots\!00}a^{26}-\frac{32\!\cdots\!61}{10\!\cdots\!20}a^{25}-\frac{13\!\cdots\!23}{78\!\cdots\!00}a^{24}+\frac{25\!\cdots\!73}{13\!\cdots\!00}a^{23}+\frac{82\!\cdots\!13}{78\!\cdots\!00}a^{22}-\frac{15\!\cdots\!11}{21\!\cdots\!44}a^{21}-\frac{31\!\cdots\!01}{78\!\cdots\!00}a^{20}+\frac{11\!\cdots\!41}{10\!\cdots\!00}a^{19}+\frac{47\!\cdots\!11}{78\!\cdots\!00}a^{18}+\frac{21\!\cdots\!79}{10\!\cdots\!00}a^{17}+\frac{15\!\cdots\!83}{13\!\cdots\!00}a^{16}+\frac{51\!\cdots\!69}{78\!\cdots\!00}a^{15}+\frac{28\!\cdots\!17}{78\!\cdots\!00}a^{14}-\frac{28\!\cdots\!43}{27\!\cdots\!00}a^{13}-\frac{46\!\cdots\!03}{78\!\cdots\!00}a^{12}+\frac{61\!\cdots\!37}{31\!\cdots\!20}a^{11}+\frac{17\!\cdots\!51}{15\!\cdots\!60}a^{10}+\frac{13\!\cdots\!37}{10\!\cdots\!00}a^{9}+\frac{26\!\cdots\!43}{39\!\cdots\!00}a^{8}+\frac{35\!\cdots\!13}{99\!\cdots\!00}a^{7}+\frac{13\!\cdots\!07}{70\!\cdots\!00}a^{6}+\frac{16\!\cdots\!17}{27\!\cdots\!80}a^{5}+\frac{13\!\cdots\!27}{48\!\cdots\!00}a^{4}+\frac{64\!\cdots\!39}{11\!\cdots\!25}a^{3}+\frac{91\!\cdots\!21}{44\!\cdots\!00}a^{2}+\frac{42\!\cdots\!11}{17\!\cdots\!05}a+\frac{69\!\cdots\!44}{12\!\cdots\!75}$
| 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: | \( 20661660535943.168 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 20661660535943.168 \cdot 336}{4\cdot\sqrt{802755791064235699686694954262517410037760000000000000000}}\cr\approx \mathstrut & 0.361435393172598 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^32 - 36*x^30 + 576*x^28 - 5474*x^26 + 34020*x^24 - 130166*x^22 + 187427*x^20 + 377425*x^18 + 1235554*x^16 - 19352440*x^14 + 35136537*x^12 + 25383639*x^10 + 8167525*x^8 + 1500436*x^6 + 160416*x^4 + 8704*x^2 + 256)
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 - 36*x^30 + 576*x^28 - 5474*x^26 + 34020*x^24 - 130166*x^22 + 187427*x^20 + 377425*x^18 + 1235554*x^16 - 19352440*x^14 + 35136537*x^12 + 25383639*x^10 + 8167525*x^8 + 1500436*x^6 + 160416*x^4 + 8704*x^2 + 256, 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 - 36*x^30 + 576*x^28 - 5474*x^26 + 34020*x^24 - 130166*x^22 + 187427*x^20 + 377425*x^18 + 1235554*x^16 - 19352440*x^14 + 35136537*x^12 + 25383639*x^10 + 8167525*x^8 + 1500436*x^6 + 160416*x^4 + 8704*x^2 + 256);
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 - 36*x^30 + 576*x^28 - 5474*x^26 + 34020*x^24 - 130166*x^22 + 187427*x^20 + 377425*x^18 + 1235554*x^16 - 19352440*x^14 + 35136537*x^12 + 25383639*x^10 + 8167525*x^8 + 1500436*x^6 + 160416*x^4 + 8704*x^2 + 256);
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 | ${\href{/padicField/3.8.0.1}{8} }^{4}$ | R | R | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{8}$ | ${\href{/padicField/13.2.0.1}{2} }^{16}$ | ${\href{/padicField/17.8.0.1}{8} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{8}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{8}$ |
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 | $x^{8} + 8 x^{7} + 32 x^{6} + 82 x^{5} + 148 x^{4} + 184 x^{3} + 137 x^{2} + 44 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
|
\(5\)
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
|
\(29\)
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
|
\(521\)
| 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 $4$ | $2$ | $2$ | $2$ |