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Properties

Label 16.0.901...000.3
Degree $16$
Signature $[0, 8]$
Discriminant $9.012\times 10^{35}$
Root discriminant \(176.68\)
Ramified primes $2,5,29$
Class number $16886916$ (GRH)
Class group [6, 2814486] (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119)
 
gp: K = bnfinit(y^16 - 8*y^15 + 484*y^14 - 3248*y^13 + 95050*y^12 - 528440*y^11 + 9696320*y^10 - 43714024*y^9 + 546280763*y^8 - 1928520760*y^7 + 16644202576*y^6 - 43360708648*y^5 + 248075557402*y^4 - 426048616864*y^3 + 1380990979420*y^2 - 1174884720024*y + 4002399015119, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119)
 

\( x^{16} - 8 x^{15} + 484 x^{14} - 3248 x^{13} + 95050 x^{12} - 528440 x^{11} + 9696320 x^{10} + \cdots + 4002399015119 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(901163823601834744225359462400000000\) \(\medspace = 2^{62}\cdot 5^{8}\cdot 29^{8}\) 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:  \(176.68\)
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:  $2^{31/8}5^{1/2}29^{1/2}\approx 176.67505464614376$
Ramified primes:   \(2\), \(5\), \(29\) 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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4640=2^{5}\cdot 5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{4640}(1,·)$, $\chi_{4640}(579,·)$, $\chi_{4640}(1161,·)$, $\chi_{4640}(1739,·)$, $\chi_{4640}(2321,·)$, $\chi_{4640}(4291,·)$, $\chi_{4640}(3481,·)$, $\chi_{4640}(2899,·)$, $\chi_{4640}(4059,·)$, $\chi_{4640}(929,·)$, $\chi_{4640}(2089,·)$, $\chi_{4640}(811,·)$, $\chi_{4640}(3249,·)$, $\chi_{4640}(1971,·)$, $\chi_{4640}(4409,·)$, $\chi_{4640}(3131,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{29}a^{4}-\frac{2}{29}a^{3}-\frac{1}{29}a^{2}+\frac{2}{29}a+\frac{1}{29}$, $\frac{1}{29}a^{5}-\frac{5}{29}a^{3}+\frac{5}{29}a+\frac{2}{29}$, $\frac{1}{29}a^{6}-\frac{10}{29}a^{3}+\frac{12}{29}a+\frac{5}{29}$, $\frac{1}{29}a^{7}+\frac{9}{29}a^{3}+\frac{2}{29}a^{2}-\frac{4}{29}a+\frac{10}{29}$, $\frac{1}{841}a^{8}-\frac{4}{841}a^{7}+\frac{2}{841}a^{6}+\frac{8}{841}a^{5}-\frac{5}{841}a^{4}-\frac{8}{841}a^{3}+\frac{2}{841}a^{2}+\frac{4}{841}a+\frac{1}{841}$, $\frac{1}{841}a^{9}-\frac{14}{841}a^{7}-\frac{13}{841}a^{6}-\frac{2}{841}a^{5}+\frac{1}{841}a^{4}+\frac{347}{841}a^{3}-\frac{17}{841}a^{2}-\frac{418}{841}a-\frac{170}{841}$, $\frac{1}{841}a^{10}-\frac{11}{841}a^{7}-\frac{3}{841}a^{6}-\frac{3}{841}a^{5}-\frac{13}{841}a^{4}+\frac{161}{841}a^{3}+\frac{16}{841}a^{2}-\frac{172}{841}a-\frac{73}{841}$, $\frac{1}{841}a^{11}+\frac{11}{841}a^{7}-\frac{10}{841}a^{6}-\frac{12}{841}a^{5}-\frac{10}{841}a^{4}-\frac{275}{841}a^{3}+\frac{82}{841}a^{2}+\frac{14}{29}a+\frac{156}{841}$, $\frac{1}{24389}a^{12}-\frac{6}{24389}a^{11}+\frac{9}{24389}a^{10}+\frac{10}{24389}a^{9}-\frac{1}{24389}a^{8}-\frac{122}{24389}a^{7}+\frac{99}{24389}a^{6}+\frac{238}{24389}a^{5}-\frac{175}{24389}a^{4}-\frac{242}{24389}a^{3}+\frac{67}{24389}a^{2}+\frac{122}{24389}a+\frac{30}{24389}$, $\frac{1}{24389}a^{13}+\frac{2}{24389}a^{11}+\frac{6}{24389}a^{10}+\frac{1}{24389}a^{9}-\frac{12}{24389}a^{8}-\frac{169}{24389}a^{7}+\frac{20}{24389}a^{6}-\frac{400}{24389}a^{5}+\frac{216}{24389}a^{4}+\frac{3139}{24389}a^{3}-\frac{172}{24389}a^{2}-\frac{3240}{24389}a-\frac{1270}{24389}$, $\frac{1}{19\!\cdots\!77}a^{14}-\frac{7}{19\!\cdots\!77}a^{13}-\frac{19\!\cdots\!39}{19\!\cdots\!77}a^{12}+\frac{11\!\cdots\!25}{19\!\cdots\!77}a^{11}-\frac{65\!\cdots\!15}{19\!\cdots\!77}a^{10}-\frac{77\!\cdots\!68}{19\!\cdots\!77}a^{9}-\frac{86\!\cdots\!14}{19\!\cdots\!77}a^{8}+\frac{23\!\cdots\!01}{19\!\cdots\!77}a^{7}-\frac{17\!\cdots\!08}{19\!\cdots\!77}a^{6}-\frac{14\!\cdots\!12}{19\!\cdots\!77}a^{5}-\frac{92\!\cdots\!28}{19\!\cdots\!77}a^{4}-\frac{11\!\cdots\!09}{19\!\cdots\!77}a^{3}+\frac{21\!\cdots\!13}{66\!\cdots\!13}a^{2}-\frac{42\!\cdots\!88}{19\!\cdots\!77}a+\frac{54\!\cdots\!77}{62\!\cdots\!67}$, $\frac{1}{95\!\cdots\!29}a^{15}+\frac{246922081}{95\!\cdots\!29}a^{14}+\frac{10\!\cdots\!63}{95\!\cdots\!29}a^{13}-\frac{11\!\cdots\!74}{95\!\cdots\!29}a^{12}+\frac{49\!\cdots\!34}{95\!\cdots\!29}a^{11}+\frac{20\!\cdots\!49}{95\!\cdots\!29}a^{10}-\frac{19\!\cdots\!43}{95\!\cdots\!29}a^{9}+\frac{26\!\cdots\!70}{95\!\cdots\!29}a^{8}-\frac{26\!\cdots\!08}{95\!\cdots\!29}a^{7}-\frac{50\!\cdots\!96}{30\!\cdots\!59}a^{6}-\frac{24\!\cdots\!15}{95\!\cdots\!29}a^{5}+\frac{66\!\cdots\!91}{95\!\cdots\!29}a^{4}+\frac{20\!\cdots\!91}{95\!\cdots\!29}a^{3}+\frac{44\!\cdots\!52}{95\!\cdots\!29}a^{2}+\frac{18\!\cdots\!70}{95\!\cdots\!29}a-\frac{48\!\cdots\!95}{10\!\cdots\!71}$ 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

$C_{6}\times C_{2814486}$, which has order $16886916$ (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:  $7$
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:   $\frac{79\!\cdots\!44}{39\!\cdots\!61}a^{15}-\frac{59\!\cdots\!80}{39\!\cdots\!61}a^{14}+\frac{36\!\cdots\!20}{39\!\cdots\!61}a^{13}-\frac{22\!\cdots\!50}{39\!\cdots\!61}a^{12}+\frac{66\!\cdots\!00}{39\!\cdots\!61}a^{11}-\frac{34\!\cdots\!36}{39\!\cdots\!61}a^{10}+\frac{62\!\cdots\!80}{39\!\cdots\!61}a^{9}-\frac{25\!\cdots\!95}{39\!\cdots\!61}a^{8}+\frac{10\!\cdots\!00}{12\!\cdots\!31}a^{7}-\frac{97\!\cdots\!00}{39\!\cdots\!61}a^{6}+\frac{79\!\cdots\!96}{39\!\cdots\!61}a^{5}-\frac{17\!\cdots\!90}{39\!\cdots\!61}a^{4}+\frac{90\!\cdots\!60}{39\!\cdots\!61}a^{3}-\frac{11\!\cdots\!00}{39\!\cdots\!61}a^{2}+\frac{23\!\cdots\!04}{39\!\cdots\!61}a+\frac{30\!\cdots\!84}{12\!\cdots\!31}$, $\frac{367920}{16\!\cdots\!69}a^{14}-\frac{2575440}{16\!\cdots\!69}a^{13}+\frac{185259506}{16\!\cdots\!69}a^{12}-\frac{1078076316}{16\!\cdots\!69}a^{11}+\frac{36654440542}{16\!\cdots\!69}a^{10}-\frac{173451217800}{16\!\cdots\!69}a^{9}+\frac{3619239738559}{16\!\cdots\!69}a^{8}-\frac{13448005261792}{16\!\cdots\!69}a^{7}+\frac{185121545009252}{16\!\cdots\!69}a^{6}-\frac{509013363373144}{16\!\cdots\!69}a^{5}+\frac{44\!\cdots\!57}{16\!\cdots\!69}a^{4}-\frac{80\!\cdots\!36}{16\!\cdots\!69}a^{3}+\frac{28\!\cdots\!96}{16\!\cdots\!69}a^{2}-\frac{24\!\cdots\!04}{16\!\cdots\!69}a+\frac{878266228234050}{53\!\cdots\!99}$, $\frac{738990235236}{22\!\cdots\!97}a^{14}-\frac{5172931646652}{22\!\cdots\!97}a^{13}+\frac{315883618075950}{22\!\cdots\!97}a^{12}-\frac{18\!\cdots\!24}{22\!\cdots\!97}a^{11}+\frac{53\!\cdots\!32}{22\!\cdots\!97}a^{10}-\frac{24\!\cdots\!46}{22\!\cdots\!97}a^{9}+\frac{44\!\cdots\!37}{22\!\cdots\!97}a^{8}-\frac{16\!\cdots\!40}{22\!\cdots\!97}a^{7}+\frac{19\!\cdots\!48}{22\!\cdots\!97}a^{6}-\frac{52\!\cdots\!24}{22\!\cdots\!97}a^{5}+\frac{39\!\cdots\!58}{22\!\cdots\!97}a^{4}-\frac{70\!\cdots\!36}{22\!\cdots\!97}a^{3}+\frac{24\!\cdots\!83}{22\!\cdots\!97}a^{2}-\frac{20\!\cdots\!22}{22\!\cdots\!97}a-\frac{37\!\cdots\!45}{74\!\cdots\!87}$, $\frac{79\!\cdots\!44}{39\!\cdots\!61}a^{15}-\frac{57\!\cdots\!60}{32\!\cdots\!01}a^{14}+\frac{31\!\cdots\!80}{32\!\cdots\!01}a^{13}-\frac{22\!\cdots\!24}{32\!\cdots\!01}a^{12}+\frac{58\!\cdots\!64}{32\!\cdots\!01}a^{11}-\frac{35\!\cdots\!94}{32\!\cdots\!01}a^{10}+\frac{55\!\cdots\!80}{32\!\cdots\!01}a^{9}-\frac{28\!\cdots\!06}{32\!\cdots\!01}a^{8}+\frac{28\!\cdots\!68}{32\!\cdots\!01}a^{7}-\frac{38\!\cdots\!68}{10\!\cdots\!71}a^{6}+\frac{76\!\cdots\!12}{32\!\cdots\!01}a^{5}-\frac{23\!\cdots\!43}{32\!\cdots\!01}a^{4}+\frac{91\!\cdots\!04}{32\!\cdots\!01}a^{3}-\frac{15\!\cdots\!84}{32\!\cdots\!01}a^{2}+\frac{64\!\cdots\!60}{85\!\cdots\!47}a+\frac{85\!\cdots\!94}{10\!\cdots\!71}$, $\frac{79\!\cdots\!44}{39\!\cdots\!61}a^{15}-\frac{15\!\cdots\!70}{95\!\cdots\!29}a^{14}+\frac{89\!\cdots\!30}{95\!\cdots\!29}a^{13}-\frac{57\!\cdots\!25}{95\!\cdots\!29}a^{12}+\frac{16\!\cdots\!00}{95\!\cdots\!29}a^{11}-\frac{84\!\cdots\!43}{95\!\cdots\!29}a^{10}+\frac{15\!\cdots\!40}{95\!\cdots\!29}a^{9}-\frac{59\!\cdots\!60}{95\!\cdots\!29}a^{8}+\frac{74\!\cdots\!00}{95\!\cdots\!29}a^{7}-\frac{70\!\cdots\!15}{32\!\cdots\!01}a^{6}+\frac{18\!\cdots\!18}{95\!\cdots\!29}a^{5}-\frac{30\!\cdots\!60}{95\!\cdots\!29}a^{4}+\frac{19\!\cdots\!60}{95\!\cdots\!29}a^{3}-\frac{20\!\cdots\!00}{95\!\cdots\!29}a^{2}+\frac{51\!\cdots\!46}{95\!\cdots\!29}a-\frac{62\!\cdots\!07}{30\!\cdots\!59}$, $\frac{148275119236350}{19\!\cdots\!77}a^{14}-\frac{10\!\cdots\!50}{19\!\cdots\!77}a^{13}+\frac{45\!\cdots\!75}{19\!\cdots\!77}a^{12}-\frac{25\!\cdots\!00}{19\!\cdots\!77}a^{11}+\frac{23\!\cdots\!07}{19\!\cdots\!77}a^{10}-\frac{94\!\cdots\!60}{19\!\cdots\!77}a^{9}-\frac{51\!\cdots\!35}{19\!\cdots\!77}a^{8}+\frac{21\!\cdots\!00}{19\!\cdots\!77}a^{7}-\frac{23\!\cdots\!05}{66\!\cdots\!13}a^{6}+\frac{19\!\cdots\!38}{19\!\cdots\!77}a^{5}-\frac{24\!\cdots\!50}{19\!\cdots\!77}a^{4}+\frac{45\!\cdots\!40}{19\!\cdots\!77}a^{3}-\frac{16\!\cdots\!00}{19\!\cdots\!77}a^{2}+\frac{14\!\cdots\!30}{19\!\cdots\!77}a-\frac{34\!\cdots\!88}{62\!\cdots\!67}$, $\frac{79\!\cdots\!44}{39\!\cdots\!61}a^{15}-\frac{13\!\cdots\!48}{11\!\cdots\!69}a^{14}+\frac{10\!\cdots\!76}{11\!\cdots\!69}a^{13}-\frac{50\!\cdots\!00}{11\!\cdots\!69}a^{12}+\frac{18\!\cdots\!52}{11\!\cdots\!69}a^{11}-\frac{72\!\cdots\!80}{11\!\cdots\!69}a^{10}+\frac{16\!\cdots\!78}{11\!\cdots\!69}a^{9}-\frac{51\!\cdots\!06}{11\!\cdots\!69}a^{8}+\frac{81\!\cdots\!20}{11\!\cdots\!69}a^{7}-\frac{18\!\cdots\!04}{11\!\cdots\!69}a^{6}+\frac{20\!\cdots\!36}{11\!\cdots\!69}a^{5}-\frac{31\!\cdots\!44}{11\!\cdots\!69}a^{4}+\frac{22\!\cdots\!68}{11\!\cdots\!69}a^{3}-\frac{22\!\cdots\!09}{11\!\cdots\!69}a^{2}+\frac{58\!\cdots\!22}{11\!\cdots\!69}a-\frac{82\!\cdots\!27}{36\!\cdots\!99}$ 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:  \( 12198.951274811623 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 12198.951274811623 \cdot 16886916}{2\cdot\sqrt{901163823601834744225359462400000000}}\cr\approx \mathstrut & 0.263560253391335 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119)
 
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^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119, 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^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119);
 
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^16 - 8*x^15 + 484*x^14 - 3248*x^13 + 95050*x^12 - 528440*x^11 + 9696320*x^10 - 43714024*x^9 + 546280763*x^8 - 1928520760*x^7 + 16644202576*x^6 - 43360708648*x^5 + 248075557402*x^4 - 426048616864*x^3 + 1380990979420*x^2 - 1174884720024*x + 4002399015119);
 
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

$C_2\times C_8$ (as 16T5):

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 16
The 16 conjugacy class representatives for $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\zeta_{16})^+\), 4.4.51200.1, 8.8.2621440000.1, 8.0.1518874382041088.26, 8.0.949296488775680000.1

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 ${\href{/padicField/3.8.0.1}{8} }^{2}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.8.0.1}{8} }^{2}$ ${\href{/padicField/13.8.0.1}{8} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.8.0.1}{8} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ R ${\href{/padicField/31.1.0.1}{1} }^{16}$ ${\href{/padicField/37.8.0.1}{8} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.8.0.1}{8} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.8.0.1}{8} }^{2}$ ${\href{/padicField/59.8.0.1}{8} }^{2}$

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 2.16.62.1$x^{16} + 16 x^{14} + 60 x^{12} + 16 x^{11} - 56 x^{10} + 144 x^{9} - 184 x^{8} + 192 x^{7} + 384 x^{6} + 448 x^{5} + 1368 x^{4} + 1376 x^{3} + 1936 x^{2} + 992 x + 1468$$8$$2$$62$$C_8\times C_2$$[3, 4, 5]^{2}$
\(5\) Copy content Toggle raw display 5.16.8.1$x^{16} + 160 x^{15} + 11240 x^{14} + 453600 x^{13} + 11536702 x^{12} + 190484240 x^{11} + 2020220586 x^{10} + 13041178608 x^{9} + 45239382035 x^{8} + 65384309200 x^{7} + 52374358166 x^{6} + 35488260768 x^{5} + 46408266743 x^{4} + 66345171264 x^{3} + 136057926318 x^{2} + 159173865296 x + 74196697609$$2$$8$$8$$C_8\times C_2$$[\ ]_{2}^{8}$
\(29\) Copy content Toggle raw display 29.8.4.2$x^{8} + 1682 x^{4} - 365835 x^{2} + 1414562$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
29.8.4.2$x^{8} + 1682 x^{4} - 365835 x^{2} + 1414562$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$