Properties

Label 32.32.187...000.2
Degree $32$
Signature $[32, 0]$
Discriminant $1.871\times 10^{74}$
Root discriminant \(209.41\)
Ramified primes $2,5$
Class number $2$ (GRH)
Class group [2] (GRH)
Galois group $C_{32}$ (as 32T33)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 28500781250)
 
gp: K = bnfinit(y^32 - 160*y^30 + 11600*y^28 - 504000*y^26 + 14625000*y^24 - 299000000*y^22 + 4427500000*y^20 - 48070000000*y^18 + 383057812500*y^16 - 2220625000000*y^14 + 9185312500000*y^12 - 26243750000000*y^10 + 49207031250000*y^8 - 55781250000000*y^6 + 33203125000000*y^4 - 7812500000000*y^2 + 28500781250, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 28500781250);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 28500781250)
 

\( x^{32} - 160 x^{30} + 11600 x^{28} - 504000 x^{26} + 14625000 x^{24} - 299000000 x^{22} + 4427500000 x^{20} - 48070000000 x^{18} + 383057812500 x^{16} + \cdots + 28500781250 \) 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:  $[32, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(187072209578355573530071658587684226515959365500928000000000000000000000000\) \(\medspace = 2^{191}\cdot 5^{24}\) 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:  \(209.41\)
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:  $2^{191/32}5^{3/4}\approx 209.41138535741914$
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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(640=2^{7}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{640}(1,·)$, $\chi_{640}(3,·)$, $\chi_{640}(9,·)$, $\chi_{640}(267,·)$, $\chi_{640}(401,·)$, $\chi_{640}(403,·)$, $\chi_{640}(409,·)$, $\chi_{640}(27,·)$, $\chi_{640}(161,·)$, $\chi_{640}(163,·)$, $\chi_{640}(169,·)$, $\chi_{640}(427,·)$, $\chi_{640}(561,·)$, $\chi_{640}(563,·)$, $\chi_{640}(569,·)$, $\chi_{640}(187,·)$, $\chi_{640}(321,·)$, $\chi_{640}(323,·)$, $\chi_{640}(329,·)$, $\chi_{640}(587,·)$, $\chi_{640}(81,·)$, $\chi_{640}(83,·)$, $\chi_{640}(89,·)$, $\chi_{640}(347,·)$, $\chi_{640}(481,·)$, $\chi_{640}(483,·)$, $\chi_{640}(489,·)$, $\chi_{640}(107,·)$, $\chi_{640}(241,·)$, $\chi_{640}(243,·)$, $\chi_{640}(249,·)$, $\chi_{640}(507,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{5}a^{6}$, $\frac{1}{5}a^{7}$, $\frac{1}{25}a^{8}$, $\frac{1}{25}a^{9}$, $\frac{1}{25}a^{10}$, $\frac{1}{25}a^{11}$, $\frac{1}{125}a^{12}$, $\frac{1}{125}a^{13}$, $\frac{1}{125}a^{14}$, $\frac{1}{125}a^{15}$, $\frac{1}{539375}a^{16}-\frac{16}{107875}a^{14}-\frac{343}{107875}a^{12}-\frac{34}{21575}a^{10}+\frac{103}{21575}a^{8}-\frac{403}{4315}a^{6}-\frac{287}{4315}a^{4}-\frac{233}{863}a^{2}+\frac{387}{863}$, $\frac{1}{103020625}a^{17}+\frac{46586}{20604125}a^{15}-\frac{33137}{20604125}a^{13}+\frac{35349}{4120825}a^{11}+\frac{63102}{4120825}a^{9}-\frac{78073}{824165}a^{7}-\frac{63286}{824165}a^{5}-\frac{66684}{164833}a^{3}+\frac{79783}{164833}a$, $\frac{1}{103020625}a^{18}-\frac{18}{20604125}a^{16}+\frac{68857}{20604125}a^{14}-\frac{6233}{20604125}a^{12}+\frac{73798}{4120825}a^{10}+\frac{3859}{4120825}a^{8}+\frac{53797}{824165}a^{6}-\frac{49212}{824165}a^{4}-\frac{21447}{164833}a^{2}-\frac{79}{863}$, $\frac{1}{103020625}a^{19}-\frac{24061}{20604125}a^{15}-\frac{21569}{20604125}a^{13}-\frac{41452}{4120825}a^{11}+\frac{78717}{4120825}a^{9}-\frac{49787}{824165}a^{7}+\frac{24203}{824165}a^{5}+\frac{75814}{164833}a^{3}+\frac{77562}{164833}a$, $\frac{1}{515103125}a^{20}+\frac{1}{20604125}a^{16}+\frac{39196}{20604125}a^{14}-\frac{10888}{4120825}a^{12}+\frac{54631}{4120825}a^{10}-\frac{43484}{4120825}a^{8}-\frac{1577}{824165}a^{6}-\frac{14595}{164833}a^{4}+\frac{45423}{164833}a^{2}-\frac{429}{863}$, $\frac{1}{515103125}a^{21}-\frac{28901}{20604125}a^{15}-\frac{53588}{20604125}a^{13}+\frac{42719}{4120825}a^{11}-\frac{29328}{4120825}a^{9}+\frac{59122}{824165}a^{7}+\frac{78622}{824165}a^{5}+\frac{49177}{164833}a^{3}+\frac{13645}{164833}a$, $\frac{1}{515103125}a^{22}+\frac{82}{103020625}a^{16}-\frac{59318}{20604125}a^{14}+\frac{70154}{20604125}a^{12}-\frac{296}{4120825}a^{10}+\frac{4687}{824165}a^{8}-\frac{778}{164833}a^{6}-\frac{42334}{824165}a^{4}-\frac{49194}{164833}a^{2}+\frac{402}{863}$, $\frac{1}{515103125}a^{23}+\frac{76622}{20604125}a^{15}-\frac{14773}{20604125}a^{13}+\frac{13616}{824165}a^{11}-\frac{41106}{4120825}a^{9}-\frac{30391}{824165}a^{7}+\frac{7459}{164833}a^{5}-\frac{20595}{164833}a^{3}-\frac{36937}{164833}a$, $\frac{1}{2575515625}a^{24}+\frac{31}{103020625}a^{16}-\frac{63196}{20604125}a^{14}-\frac{34487}{20604125}a^{12}+\frac{24478}{4120825}a^{10}-\frac{1456}{824165}a^{8}+\frac{49861}{824165}a^{6}+\frac{38233}{824165}a^{4}+\frac{3385}{164833}a^{2}+\frac{153}{863}$, $\frac{1}{2575515625}a^{25}-\frac{4773}{4120825}a^{15}+\frac{3762}{20604125}a^{13}-\frac{82343}{4120825}a^{11}+\frac{14554}{4120825}a^{9}-\frac{2371}{824165}a^{7}+\frac{22103}{824165}a^{5}-\frac{72240}{164833}a^{3}+\frac{28445}{164833}a$, $\frac{1}{2575515625}a^{26}+\frac{2}{4120825}a^{16}+\frac{71758}{20604125}a^{14}+\frac{15743}{20604125}a^{12}+\frac{76629}{4120825}a^{10}-\frac{15741}{824165}a^{8}+\frac{45214}{824165}a^{6}-\frac{1379}{164833}a^{4}+\frac{70847}{164833}a^{2}+\frac{235}{863}$, $\frac{1}{2575515625}a^{27}+\frac{10024}{4120825}a^{15}+\frac{24263}{20604125}a^{13}-\frac{42491}{4120825}a^{11}+\frac{12571}{824165}a^{9}-\frac{7128}{824165}a^{7}+\frac{25578}{824165}a^{5}-\frac{56446}{164833}a^{3}+\frac{11727}{164833}a$, $\frac{1}{12877578125}a^{28}+\frac{78}{103020625}a^{16}-\frac{51607}{20604125}a^{14}-\frac{20717}{20604125}a^{12}+\frac{65669}{4120825}a^{10}-\frac{51631}{4120825}a^{8}+\frac{29449}{824165}a^{6}-\frac{34863}{824165}a^{4}-\frac{8045}{164833}a^{2}-\frac{423}{863}$, $\frac{1}{12877578125}a^{29}-\frac{58989}{20604125}a^{15}-\frac{73359}{20604125}a^{13}-\frac{2169}{164833}a^{11}-\frac{28597}{4120825}a^{9}+\frac{20322}{824165}a^{7}-\frac{8709}{164833}a^{5}-\frac{81349}{164833}a^{3}-\frac{40213}{164833}a$, $\frac{1}{12877578125}a^{30}-\frac{41}{103020625}a^{16}-\frac{11666}{20604125}a^{14}-\frac{50902}{20604125}a^{12}-\frac{104}{824165}a^{10}-\frac{17383}{4120825}a^{8}-\frac{45264}{824165}a^{6}+\frac{1927}{164833}a^{4}-\frac{17484}{164833}a^{2}+\frac{332}{863}$, $\frac{1}{12877578125}a^{31}-\frac{79636}{20604125}a^{15}+\frac{73978}{20604125}a^{13}-\frac{34708}{4120825}a^{11}-\frac{67529}{4120825}a^{9}+\frac{50403}{824165}a^{7}+\frac{52237}{824165}a^{5}+\frac{50633}{164833}a^{3}+\frac{37855}{164833}a$ 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

$C_{2}$, which has order $2$ (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:  $31$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{1}{539375}a^{16}-\frac{16}{107875}a^{14}+\frac{104}{21575}a^{12}-\frac{352}{4315}a^{10}+\frac{660}{863}a^{8}-\frac{3360}{863}a^{6}+\frac{8400}{863}a^{4}-\frac{8000}{863}a^{2}+\frac{2113}{863}$, $\frac{7}{2575515625}a^{24}-\frac{168}{515103125}a^{22}+\frac{1764}{103020625}a^{20}-\frac{2128}{4120825}a^{18}+\frac{40698}{4120825}a^{16}-\frac{102816}{824165}a^{14}+\frac{173264}{164833}a^{12}-\frac{960960}{164833}a^{10}+\frac{84449079}{4120825}a^{8}-\frac{34952632}{824165}a^{6}+\frac{7301580}{164833}a^{4}-\frac{2326320}{164833}a^{2}-\frac{2413}{863}$, $\frac{24}{2575515625}a^{24}-\frac{576}{515103125}a^{22}+\frac{6048}{103020625}a^{20}-\frac{7296}{4120825}a^{18}+\frac{139536}{4120825}a^{16}-\frac{352512}{824165}a^{14}+\frac{594048}{164833}a^{12}-\frac{3294720}{164833}a^{10}+\frac{289563247}{4120825}a^{8}-\frac{120025976}{824165}a^{6}+\frac{25504940}{164833}a^{4}-\frac{9859760}{164833}a^{2}-\frac{13}{863}$, $\frac{4}{12877578125}a^{28}-\frac{112}{2575515625}a^{26}+\frac{56}{20604125}a^{24}-\frac{10304}{103020625}a^{22}+\frac{49588}{20604125}a^{20}-\frac{163856}{4120825}a^{18}+\frac{379848}{824165}a^{16}-\frac{620160}{164833}a^{14}+\frac{3527160}{164833}a^{12}-\frac{13613600}{164833}a^{10}+\frac{34034000}{164833}a^{8}-\frac{50960000}{164833}a^{6}+\frac{198986057}{824165}a^{4}-\frac{11944228}{164833}a^{2}-\frac{1593}{863}$, $\frac{44}{515103125}a^{20}-\frac{176}{20604125}a^{18}+\frac{1496}{4120825}a^{16}-\frac{1408}{164833}a^{14}+\frac{2502737}{20604125}a^{12}-\frac{4407244}{4120825}a^{10}+\frac{4731798}{824165}a^{8}-\frac{2930544}{164833}a^{6}+\frac{4661925}{164833}a^{4}-\frac{2963300}{164833}a^{2}+\frac{2613}{863}$, $\frac{4}{12877578125}a^{28}-\frac{112}{2575515625}a^{26}+\frac{6993}{2575515625}a^{24}-\frac{51352}{515103125}a^{22}+\frac{1230763}{515103125}a^{20}-\frac{808172}{20604125}a^{18}+\frac{1854564}{4120825}a^{16}-\frac{2979264}{824165}a^{14}+\frac{412585741}{20604125}a^{12}-\frac{304641692}{4120825}a^{10}+\frac{704500991}{4120825}a^{8}-\frac{182982328}{824165}a^{6}+\frac{110329532}{824165}a^{4}-\frac{5109808}{164833}a^{2}+\frac{207}{863}$, $\frac{117}{515103125}a^{20}-\frac{468}{20604125}a^{18}+\frac{3978}{4120825}a^{16}-\frac{3744}{164833}a^{14}+\frac{6651259}{20604125}a^{12}-\frac{11674308}{4120825}a^{10}+\frac{12379986}{824165}a^{8}-\frac{7373008}{164833}a^{6}+\frac{10429725}{164833}a^{4}-\frac{4508100}{164833}a^{2}+\frac{613}{863}$, $\frac{3}{12877578125}a^{28}-\frac{84}{2575515625}a^{26}+\frac{42}{20604125}a^{24}-\frac{7728}{103020625}a^{22}+\frac{37191}{20604125}a^{20}-\frac{3072271}{103020625}a^{18}+\frac{7121628}{20604125}a^{16}-\frac{58121143}{20604125}a^{14}+\frac{66065132}{4120825}a^{12}-\frac{50919671}{824165}a^{10}+\frac{25417890}{164833}a^{8}-\frac{38270610}{164833}a^{6}+\frac{157084251}{824165}a^{4}-\frac{12052629}{164833}a^{2}+\frac{7949}{863}$, $\frac{2}{2575515625}a^{26}-\frac{52}{515103125}a^{24}+\frac{598}{103020625}a^{22}-\frac{100056}{515103125}a^{20}+\frac{86274}{20604125}a^{18}-\frac{250444}{4120825}a^{16}+\frac{99368}{164833}a^{14}-\frac{83687263}{20604125}a^{12}+\frac{74600256}{4120825}a^{10}-\frac{41743202}{824165}a^{8}+\frac{66687369}{824165}a^{6}-\frac{10178609}{164833}a^{4}+\frac{2715705}{164833}a^{2}+\frac{389}{863}$, $\frac{2}{12877578125}a^{30}-\frac{304}{12877578125}a^{28}+\frac{4162}{2575515625}a^{26}-\frac{1356}{20604125}a^{24}+\frac{182804}{103020625}a^{22}-\frac{687148}{20604125}a^{20}+\frac{1846306}{4120825}a^{18}-\frac{3577548}{824165}a^{16}+\frac{4980660}{164833}a^{14}-\frac{24522160}{164833}a^{12}+\frac{82897100}{164833}a^{10}-\frac{183209000}{164833}a^{8}+\frac{244335000}{164833}a^{6}-\frac{855236057}{824165}a^{4}+\frac{46991787}{164833}a^{2}-\frac{4349}{863}$, $\frac{41}{515103125}a^{22}-\frac{23}{2696875}a^{20}+\frac{8101}{20604125}a^{18}-\frac{42024}{4120825}a^{16}+\frac{26924}{164833}a^{14}-\frac{180051}{107875}a^{12}+\frac{1790177}{164833}a^{10}-\frac{35963344}{824165}a^{8}+\frac{16719397}{164833}a^{6}-\frac{19916025}{164833}a^{4}+\frac{9343775}{164833}a^{2}-\frac{6089}{863}$, $\frac{3}{12877578125}a^{28}-\frac{84}{2575515625}a^{26}+\frac{42}{20604125}a^{24}-\frac{7728}{103020625}a^{22}+\frac{37191}{20604125}a^{20}-\frac{3072022}{103020625}a^{18}+\frac{7117146}{20604125}a^{16}-\frac{57953549}{20604125}a^{14}+\frac{65392096}{4120825}a^{12}-\frac{49354393}{824165}a^{10}+\frac{23300310}{164833}a^{8}-\frac{30349830}{164833}a^{6}+\frac{84831251}{824165}a^{4}-\frac{2393129}{164833}a^{2}-\frac{2229}{863}$, $\frac{11}{2575515625}a^{26}-\frac{286}{515103125}a^{24}+\frac{3289}{103020625}a^{22}-\frac{638}{596875}a^{20}+\frac{475651}{20604125}a^{18}-\frac{1387166}{4120825}a^{16}+\frac{555676}{164833}a^{14}-\frac{552199}{23875}a^{12}+\frac{438948494}{4120825}a^{10}-\frac{260344298}{824165}a^{8}+\frac{461940793}{824165}a^{6}-\frac{85790363}{164833}a^{4}+\frac{30489085}{164833}a^{2}+\frac{689}{863}$, $\frac{1}{12877578125}a^{30}-\frac{146}{12877578125}a^{28}+\frac{1913}{2575515625}a^{26}-\frac{594}{20604125}a^{24}+\frac{75946}{103020625}a^{22}-\frac{269192}{20604125}a^{20}+\frac{677369}{4120825}a^{18}-\frac{1219002}{824165}a^{16}+\frac{1560090}{164833}a^{14}-\frac{6970340}{164833}a^{12}+\frac{21028150}{164833}a^{10}-\frac{40553500}{164833}a^{8}+\frac{45727500}{164833}a^{6}-\frac{129138943}{824165}a^{4}+\frac{5497135}{164833}a^{2}-\frac{249}{863}$, $\frac{38}{515103125}a^{22}-\frac{4297}{515103125}a^{20}+\frac{1682}{4120825}a^{18}-\frac{46614}{4120825}a^{16}+\frac{32168}{164833}a^{14}-\frac{44689259}{20604125}a^{12}+\frac{63970089}{4120825}a^{10}-\frac{57145796}{824165}a^{8}+\frac{29561843}{164833}a^{6}-\frac{37549975}{164833}a^{4}+\frac{14833725}{164833}a^{2}-\frac{89}{863}$, $\frac{4}{12877578125}a^{29}-\frac{116}{2575515625}a^{27}+\frac{9}{2575515625}a^{26}+\frac{1508}{515103125}a^{25}-\frac{234}{515103125}a^{24}-\frac{57993}{515103125}a^{23}+\frac{2691}{103020625}a^{22}+\frac{293319}{103020625}a^{21}-\frac{450494}{515103125}a^{20}-\frac{205114}{4120825}a^{19}+\frac{389201}{20604125}a^{18}+\frac{2536443}{4120825}a^{17}-\frac{1135226}{4120825}a^{16}-\frac{4463316}{824165}a^{15}+\frac{454900}{164833}a^{14}+\frac{5543564}{164833}a^{13}-\frac{390357737}{20604125}a^{12}-\frac{23768160}{164833}a^{11}+\frac{359940994}{4120825}a^{10}+\frac{1690517829}{4120825}a^{9}-\frac{213869298}{824165}a^{8}-\frac{597639211}{824165}a^{7}+\frac{380600704}{824165}a^{6}+\frac{117282633}{164833}a^{5}-\frac{70949829}{164833}a^{4}-\frac{50900793}{164833}a^{3}+\frac{24810080}{164833}a^{2}+\frac{5376920}{164833}a+\frac{1187}{863}$, $\frac{1}{12877578125}a^{31}+\frac{3}{12877578125}a^{30}-\frac{151}{12877578125}a^{29}-\frac{446}{12877578125}a^{28}+\frac{2054}{2575515625}a^{27}+\frac{5963}{2575515625}a^{26}-\frac{16627}{515103125}a^{25}-\frac{1894}{20604125}a^{24}+\frac{3566}{4120825}a^{23}+\frac{248446}{103020625}a^{22}-\frac{333454}{20604125}a^{21}-\frac{906752}{20604125}a^{20}+\frac{892584}{4120825}a^{19}+\frac{2359819}{4120825}a^{18}-\frac{1726131}{824165}a^{17}-\frac{4416702}{824165}a^{16}+\frac{2405058}{164833}a^{15}+\frac{5920590}{164833}a^{14}-\frac{11902550}{164833}a^{13}-\frac{27965340}{164833}a^{12}+\frac{40730300}{164833}a^{11}+\frac{90311650}{164833}a^{10}-\frac{92212250}{164833}a^{9}-\frac{189728500}{164833}a^{8}+\frac{128732500}{164833}a^{7}+\frac{239102500}{164833}a^{6}-\frac{98612500}{164833}a^{5}-\frac{785388943}{824165}a^{4}+\frac{33361057}{164833}a^{3}+\frac{40544694}{164833}a^{2}-\frac{2254374}{164833}a+\frac{713}{863}$, $\frac{3}{12877578125}a^{30}-\frac{2}{12877578125}a^{29}-\frac{446}{12877578125}a^{28}+\frac{42}{2575515625}a^{27}+\frac{5963}{2575515625}a^{26}-\frac{333}{515103125}a^{25}-\frac{236637}{2575515625}a^{24}+\frac{4448}{515103125}a^{23}+\frac{1239714}{515103125}a^{22}+\frac{20456}{103020625}a^{21}-\frac{22547732}{515103125}a^{20}-\frac{1117084}{103020625}a^{19}+\frac{58336286}{103020625}a^{18}+\frac{23272049}{103020625}a^{17}-\frac{108185558}{20604125}a^{16}-\frac{57676018}{20604125}a^{15}+\frac{715746556}{20604125}a^{14}+\frac{462916412}{20604125}a^{13}-\frac{3324784969}{20604125}a^{12}-\frac{488980906}{4120825}a^{11}+\frac{2107232657}{4120825}a^{10}+\frac{1670008321}{4120825}a^{9}-\frac{4352521524}{4120825}a^{8}-\frac{703445522}{824165}a^{7}+\frac{1090291367}{824165}a^{6}+\frac{834932929}{824165}a^{5}-\frac{730235218}{824165}a^{4}-\frac{94114450}{164833}a^{3}+\frac{39542272}{164833}a^{2}+\frac{18075900}{164833}a+\frac{287}{863}$, $\frac{4}{2575515625}a^{27}-\frac{108}{515103125}a^{25}+\frac{1296}{103020625}a^{23}-\frac{3}{515103125}a^{22}-\frac{9108}{20604125}a^{21}+\frac{447}{515103125}a^{20}+\frac{8316}{824165}a^{19}-\frac{219}{4120825}a^{18}-\frac{16160027}{103020625}a^{17}+\frac{7344}{4120825}a^{16}+\frac{6978659}{4120825}a^{15}-\frac{5988}{164833}a^{14}-\frac{52393753}{4120825}a^{13}+\frac{9654259}{20604125}a^{12}+\frac{53966174}{824165}a^{11}-\frac{3161452}{824165}a^{10}-\frac{37050145}{164833}a^{9}+\frac{15957506}{824165}a^{8}+\frac{80932470}{164833}a^{7}-\frac{9276578}{164833}a^{6}-\frac{523674693}{824165}a^{5}+\frac{13655225}{164833}a^{4}+\frac{70888215}{164833}a^{3}-\frac{8034350}{164833}a^{2}-\frac{19097950}{164833}a+\frac{6113}{863}$, $\frac{18}{12877578125}a^{28}+\frac{2}{2575515625}a^{27}-\frac{504}{2575515625}a^{26}-\frac{54}{515103125}a^{25}+\frac{31476}{2575515625}a^{24}+\frac{127}{20604125}a^{23}-\frac{46261}{103020625}a^{22}-\frac{851}{4120825}a^{21}+\frac{1110584}{103020625}a^{20}+\frac{445234}{103020625}a^{19}-\frac{3658688}{20604125}a^{18}-\frac{1194321}{20604125}a^{17}+\frac{8450148}{4120825}a^{16}+\frac{1999068}{4120825}a^{15}-\frac{343320204}{20604125}a^{14}-\frac{45758982}{20604125}a^{13}+\frac{388005156}{4120825}a^{12}+\frac{8408166}{4120825}a^{11}-\frac{1480798273}{4120825}a^{10}+\frac{130792938}{4120825}a^{9}+\frac{3616779403}{4120825}a^{8}-\frac{129801582}{824165}a^{7}-\frac{1028730449}{824165}a^{6}+\frac{45879095}{164833}a^{5}+\frac{718736556}{824165}a^{4}-\frac{25293875}{164833}a^{3}-\frac{35610889}{164833}a^{2}+\frac{1676592}{164833}a+\frac{245}{863}$, $\frac{2}{12877578125}a^{30}+\frac{2}{2575515625}a^{29}-\frac{297}{12877578125}a^{28}-\frac{294}{2575515625}a^{27}+\frac{791}{515103125}a^{26}+\frac{3878}{515103125}a^{25}-\frac{155813}{2575515625}a^{24}-\frac{151487}{515103125}a^{23}+\frac{807247}{515103125}a^{22}+\frac{3896961}{515103125}a^{21}-\frac{578003}{20604125}a^{20}-\frac{13886376}{103020625}a^{19}+\frac{36563302}{103020625}a^{18}+\frac{35089997}{20604125}a^{17}-\frac{65735396}{20604125}a^{16}-\frac{316530489}{20604125}a^{15}+\frac{435919}{21575}a^{14}+\frac{81008199}{824165}a^{13}-\frac{363518077}{4120825}a^{12}-\frac{1802608887}{4120825}a^{11}+\frac{211322579}{824165}a^{10}+\frac{5391978581}{4120825}a^{9}-\frac{1955590121}{4120825}a^{8}-\frac{2055827179}{824165}a^{7}+\frac{89150529}{164833}a^{6}+\frac{2307469753}{824165}a^{5}-\frac{318069784}{824165}a^{4}-\frac{269051181}{164833}a^{3}+\frac{27278325}{164833}a^{2}+\frac{62750332}{164833}a-\frac{21595}{863}$, $\frac{2}{2575515625}a^{29}-\frac{3}{12877578125}a^{28}-\frac{293}{2575515625}a^{27}+\frac{73}{2575515625}a^{26}+\frac{19279}{2575515625}a^{25}-\frac{148}{103020625}a^{24}-\frac{30092}{103020625}a^{23}+\frac{19277}{515103125}a^{22}+\frac{774417}{103020625}a^{21}-\frac{44769}{103020625}a^{20}-\frac{13825429}{103020625}a^{19}-\frac{51401}{20604125}a^{18}+\frac{175320169}{103020625}a^{17}+\frac{17558891}{103020625}a^{16}-\frac{317923801}{20604125}a^{15}-\frac{58912293}{20604125}a^{14}+\frac{2046161818}{20604125}a^{13}+\frac{111162132}{4120825}a^{12}-\frac{365960928}{824165}a^{11}-\frac{650927936}{4120825}a^{10}+\frac{1092554711}{824165}a^{9}+\frac{472994207}{824165}a^{8}-\frac{2034529163}{824165}a^{7}-\frac{1010327878}{824165}a^{6}+\frac{2089876929}{824165}a^{5}+\frac{1117664689}{824165}a^{4}-\frac{181640596}{164833}a^{3}-\frac{92144331}{164833}a^{2}+\frac{14802740}{164833}a+\frac{6225}{863}$, $\frac{2}{12877578125}a^{31}+\frac{1}{12877578125}a^{30}-\frac{62}{2575515625}a^{29}-\frac{6}{515103125}a^{28}+\frac{868}{515103125}a^{27}+\frac{81}{103020625}a^{26}-\frac{181374}{2575515625}a^{25}-\frac{16257}{515103125}a^{24}+\frac{1008141}{515103125}a^{23}+\frac{432207}{515103125}a^{22}-\frac{19645259}{515103125}a^{21}-\frac{8026353}{515103125}a^{20}+\frac{11031806}{20604125}a^{19}+\frac{21416272}{103020625}a^{18}-\frac{564470261}{103020625}a^{17}-\frac{208100689}{103020625}a^{16}+\frac{842692039}{20604125}a^{15}+\frac{295665243}{20604125}a^{14}-\frac{4546638182}{20604125}a^{13}-\frac{1527133151}{20604125}a^{12}+\frac{3478024954}{4120825}a^{11}+\frac{1126690542}{4120825}a^{10}-\frac{365134598}{164833}a^{9}-\frac{571125293}{824165}a^{8}+\frac{3120228788}{824165}a^{7}+\frac{184225269}{164833}a^{6}-\frac{636559241}{164833}a^{5}-\frac{161795125}{164833}a^{4}+\frac{330889260}{164833}a^{3}+\frac{50810113}{164833}a^{2}-\frac{62745480}{164833}a+\frac{18635}{863}$, $\frac{1}{12877578125}a^{31}-\frac{31}{2575515625}a^{29}+\frac{434}{515103125}a^{27}-\frac{3627}{103020625}a^{25}+\frac{806}{824165}a^{23}-\frac{79}{515103125}a^{22}-\frac{9803633}{515103125}a^{21}+\frac{8807}{515103125}a^{20}+\frac{27448043}{103020625}a^{19}-\frac{16979}{20604125}a^{18}-\frac{55859262}{20604125}a^{17}+\frac{92616}{4120825}a^{16}+\frac{82495866}{4120825}a^{15}-\frac{62836}{164833}a^{14}-\frac{17453954}{164833}a^{13}+\frac{85730259}{20604125}a^{12}+\frac{1610340359}{4120825}a^{11}-\frac{120398822}{4120825}a^{10}-\frac{793026499}{824165}a^{9}+\frac{105489126}{824165}a^{8}+\frac{241717496}{164833}a^{7}-\frac{53654248}{164833}a^{6}-\frac{198957465}{164833}a^{5}+\frac{355475}{863}a^{4}+\frac{64437375}{164833}a^{3}-\frac{28685600}{164833}a^{2}-\frac{500894}{164833}a-\frac{113}{863}$, $\frac{2}{2575515625}a^{29}+\frac{3}{2575515625}a^{28}-\frac{58}{515103125}a^{27}-\frac{409}{2575515625}a^{26}+\frac{3781}{515103125}a^{25}+\frac{24844}{2575515625}a^{24}-\frac{1171}{4120825}a^{23}-\frac{177296}{515103125}a^{22}+\frac{3744242}{515103125}a^{21}+\frac{4124832}{515103125}a^{20}-\frac{13323501}{103020625}a^{19}-\frac{13132261}{103020625}a^{18}+\frac{169013206}{103020625}a^{17}+\frac{29203098}{20604125}a^{16}-\frac{308254099}{20604125}a^{15}-\frac{227405943}{20604125}a^{14}+\frac{2011320208}{20604125}a^{13}+\frac{1224837919}{20604125}a^{12}-\frac{369132634}{824165}a^{11}-\frac{886956584}{4120825}a^{10}+\frac{1152817221}{824165}a^{9}+\frac{2055456477}{4120825}a^{8}-\frac{2321674637}{824165}a^{7}-\frac{563929668}{824165}a^{6}+\frac{551422604}{164833}a^{5}+\frac{81752928}{164833}a^{4}-\frac{330313781}{164833}a^{3}-\frac{28587345}{164833}a^{2}+\frac{74551798}{164833}a+\frac{24315}{863}$, $\frac{7}{12877578125}a^{29}-\frac{203}{2575515625}a^{27}+\frac{13}{2575515625}a^{26}+\frac{2639}{515103125}a^{25}-\frac{1683}{2575515625}a^{24}-\frac{101538}{515103125}a^{23}+\frac{19267}{515103125}a^{22}+\frac{2572364}{515103125}a^{21}-\frac{642006}{515103125}a^{20}-\frac{9032449}{103020625}a^{19}+\frac{2759667}{103020625}a^{18}+\frac{22531581}{20604125}a^{17}-\frac{8009476}{20604125}a^{16}-\frac{40274768}{4120825}a^{15}+\frac{79827649}{20604125}a^{14}+\frac{10277246}{164833}a^{13}-\frac{544877178}{20604125}a^{12}-\frac{1151336313}{4120825}a^{11}+\frac{498670691}{4120825}a^{10}+\frac{702691562}{824165}a^{9}-\frac{1464994631}{4120825}a^{8}-\frac{276893843}{164833}a^{7}+\frac{102587112}{164833}a^{6}+\frac{321242820}{164833}a^{5}-\frac{94425192}{164833}a^{4}-\frac{186309067}{164833}a^{3}+\frac{36001795}{164833}a^{2}+\frac{40743380}{164833}a-\frac{15715}{863}$, $\frac{8}{12877578125}a^{29}-\frac{232}{2575515625}a^{27}-\frac{3}{515103125}a^{26}+\frac{3016}{515103125}a^{25}+\frac{78}{103020625}a^{24}-\frac{116031}{515103125}a^{23}-\frac{897}{20604125}a^{22}+\frac{587673}{103020625}a^{21}+\frac{750794}{515103125}a^{20}-\frac{412298}{4120825}a^{19}-\frac{648551}{20604125}a^{18}+\frac{5131881}{4120825}a^{17}+\frac{1891046}{4120825}a^{16}-\frac{9137772}{824165}a^{15}-\frac{757228}{164833}a^{14}+\frac{11579788}{164833}a^{13}+\frac{648927737}{20604125}a^{12}-\frac{51303720}{164833}a^{11}-\frac{596963494}{4120825}a^{10}+\frac{3843593924}{4120825}a^{9}+\frac{353294298}{824165}a^{8}-\frac{1473171566}{824165}a^{7}-\frac{624620971}{824165}a^{6}+\frac{328219698}{164833}a^{5}+\frac{115471431}{164833}a^{4}-\frac{179353986}{164833}a^{3}-\frac{41847095}{164833}a^{2}+\frac{36519940}{164833}a+\frac{12389}{863}$, $\frac{3}{12877578125}a^{28}-\frac{3}{2575515625}a^{27}-\frac{84}{2575515625}a^{26}+\frac{81}{515103125}a^{25}+\frac{42}{20604125}a^{24}-\frac{4867}{515103125}a^{23}-\frac{7728}{103020625}a^{22}+\frac{34316}{103020625}a^{21}+\frac{37191}{20604125}a^{20}-\frac{31507}{4120825}a^{19}-\frac{3071993}{103020625}a^{18}+\frac{493962}{4120825}a^{17}+\frac{7116624}{20604125}a^{16}-\frac{1079364}{824165}a^{15}-\frac{57934692}{20604125}a^{14}+\frac{1646416}{164833}a^{13}+\frac{65322978}{4120825}a^{12}-\frac{8641490}{164833}a^{11}-\frac{49223064}{824165}a^{10}+\frac{755675921}{4120825}a^{9}+\frac{23192700}{164833}a^{8}-\frac{332835789}{824165}a^{7}-\frac{30400440}{164833}a^{6}+\frac{418868584}{824165}a^{5}+\frac{92634751}{824165}a^{4}-\frac{51084395}{164833}a^{3}-\frac{5322754}{164833}a^{2}+\frac{11563450}{164833}a+\frac{4973}{863}$, $\frac{3}{12877578125}a^{30}+\frac{2}{12877578125}a^{29}-\frac{431}{12877578125}a^{28}-\frac{61}{2575515625}a^{27}+\frac{5567}{2575515625}a^{26}+\frac{167}{103020625}a^{25}-\frac{213448}{2575515625}a^{24}-\frac{6772}{103020625}a^{23}+\frac{1080899}{515103125}a^{22}+\frac{904446}{515103125}a^{21}-\frac{19016763}{515103125}a^{20}-\frac{3346878}{103020625}a^{19}+\frac{47626107}{103020625}a^{18}+\frac{43904492}{103020625}a^{17}-\frac{427489889}{103020625}a^{16}-\frac{82166192}{20604125}a^{15}+\frac{546473102}{20604125}a^{14}+\frac{544089614}{20604125}a^{13}-\frac{2436389517}{20604125}a^{12}-\frac{498528417}{4120825}a^{11}+\frac{1458021589}{4120825}a^{10}+\frac{60637139}{164833}a^{9}-\frac{2739133742}{4120825}a^{8}-\frac{113954604}{164833}a^{7}+\frac{572088846}{824165}a^{6}+\frac{580044824}{824165}a^{5}-\frac{253108396}{824165}a^{4}-\frac{46855608}{164833}a^{3}+\frac{3635416}{164833}a^{2}+\frac{1925754}{164833}a+\frac{533}{863}$, $\frac{3}{12877578125}a^{30}-\frac{446}{12877578125}a^{28}-\frac{4}{2575515625}a^{27}+\frac{5963}{2575515625}a^{26}+\frac{526}{2575515625}a^{25}-\frac{1894}{20604125}a^{24}-\frac{6096}{515103125}a^{23}+\frac{248446}{103020625}a^{22}+\frac{204657}{515103125}a^{21}-\frac{906752}{20604125}a^{20}-\frac{880357}{103020625}a^{19}+\frac{58995168}{103020625}a^{18}+\frac{2532393}{20604125}a^{17}-\frac{110411833}{20604125}a^{16}-\frac{4933944}{4120825}a^{15}+\frac{739853162}{20604125}a^{14}+\frac{1289804}{164833}a^{13}-\frac{698222908}{4120825}a^{12}-\frac{136775141}{4120825}a^{11}+\frac{449394154}{824165}a^{10}+\frac{352002091}{4120825}a^{9}-\frac{186765400}{164833}a^{8}-\frac{95948202}{824165}a^{7}+\frac{228074140}{164833}a^{6}+\frac{49250008}{824165}a^{5}-\frac{688632943}{824165}a^{4}+\frac{71200}{164833}a^{3}+\frac{29269277}{164833}a^{2}+\frac{364450}{164833}a-\frac{715}{863}$, $\frac{1}{12877578125}a^{30}+\frac{1}{2575515625}a^{29}-\frac{126}{12877578125}a^{28}-\frac{129}{2575515625}a^{27}+\frac{1386}{2575515625}a^{26}+\frac{7327}{2575515625}a^{25}-\frac{43296}{2575515625}a^{24}-\frac{48051}{515103125}a^{23}+\frac{165729}{515103125}a^{22}+\frac{1000054}{515103125}a^{21}-\frac{1888124}{515103125}a^{20}-\frac{2722539}{103020625}a^{19}+\frac{1844377}{103020625}a^{18}+\frac{23763439}{103020625}a^{17}+\frac{13635343}{103020625}a^{16}-\frac{23690547}{20604125}a^{15}-\frac{64965436}{20604125}a^{14}+\frac{29592888}{20604125}a^{13}+\frac{569390334}{20604125}a^{12}+\frac{79766982}{4120825}a^{11}-\frac{580763131}{4120825}a^{10}-\frac{106932827}{824165}a^{9}+\frac{1806535969}{4120825}a^{8}+\frac{295067302}{824165}a^{7}-\frac{653440908}{824165}a^{6}-\frac{371454917}{824165}a^{5}+\frac{601641172}{824165}a^{4}+\frac{31888413}{164833}a^{3}-\frac{40153328}{164833}a^{2}-\frac{1605802}{164833}a+\frac{279}{863}$ 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:  \( 901899884176709700000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{32}\cdot(2\pi)^{0}\cdot 901899884176709700000000000 \cdot 2}{2\cdot\sqrt{187072209578355573530071658587684226515959365500928000000000000000000000000}}\cr\approx \mathstrut & 0.283213241326724 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 28500781250)
 
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 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 28500781250, 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 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 28500781250);
 
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 - 160*x^30 + 11600*x^28 - 504000*x^26 + 14625000*x^24 - 299000000*x^22 + 4427500000*x^20 - 48070000000*x^18 + 383057812500*x^16 - 2220625000000*x^14 + 9185312500000*x^12 - 26243750000000*x^10 + 49207031250000*x^8 - 55781250000000*x^6 + 33203125000000*x^4 - 7812500000000*x^2 + 28500781250);
 
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

$C_{32}$ (as 32T33):

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 cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), 16.16.236118324143482260684800000000.1

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 $32$ R $16^{2}$ $32$ $32$ ${\href{/padicField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/padicField/47.8.0.1}{8} }^{4}$ $32$ $32$

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$$32$$1$$191$
\(5\) Copy content Toggle raw display Deg $32$$4$$8$$24$