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Properties

Label 16.0.172...761.1
Degree $16$
Signature $[0, 8]$
Discriminant $1.723\times 10^{33}$
Root discriminant \(119.47\)
Ramified primes $17,89$
Class number $1177813$ (GRH)
Class group [301, 3913] (GRH)
Galois group $C_2^2 : C_4$ (as 16T10)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856)
 
gp: K = bnfinit(y^16 - 2*y^15 + 53*y^14 - 404*y^13 + 3150*y^12 - 16220*y^11 + 43602*y^10 - 147216*y^9 + 562581*y^8 - 1745730*y^7 + 3929497*y^6 - 4677356*y^5 + 6836220*y^4 - 10615248*y^3 + 18312704*y^2 - 3638272*y + 7929856, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856)
 

\( x^{16} - 2 x^{15} + 53 x^{14} - 404 x^{13} + 3150 x^{12} - 16220 x^{11} + 43602 x^{10} - 147216 x^{9} + \cdots + 7929856 \) 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:   \(1722945145525970304337872614249761\) \(\medspace = 17^{8}\cdot 89^{12}\) 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:  \(119.47\)
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:  $17^{1/2}89^{3/4}\approx 119.47222879314523$
Ramified primes:   \(17\), \(89\) 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 over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{128}$

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{4}-\frac{1}{4}a^{3}-\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{8}a^{5}-\frac{1}{8}a^{3}$, $\frac{1}{16}a^{6}-\frac{1}{16}a^{5}-\frac{1}{16}a^{4}-\frac{3}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{16}a^{7}+\frac{3}{16}a^{3}-\frac{1}{4}a$, $\frac{1}{128}a^{8}-\frac{1}{32}a^{7}+\frac{1}{64}a^{6}-\frac{1}{16}a^{5}+\frac{1}{128}a^{4}-\frac{1}{32}a^{3}-\frac{1}{32}a^{2}+\frac{1}{8}a$, $\frac{1}{128}a^{9}+\frac{1}{64}a^{7}+\frac{1}{128}a^{5}-\frac{1}{32}a^{3}$, $\frac{1}{256}a^{10}-\frac{1}{256}a^{9}+\frac{3}{128}a^{7}-\frac{3}{256}a^{6}+\frac{15}{256}a^{5}-\frac{3}{128}a^{4}+\frac{3}{64}a^{3}+\frac{1}{32}a^{2}-\frac{1}{8}a$, $\frac{1}{1024}a^{11}-\frac{3}{1024}a^{9}-\frac{1}{256}a^{8}+\frac{7}{1024}a^{7}+\frac{3}{128}a^{6}-\frac{41}{1024}a^{5}+\frac{15}{256}a^{4}+\frac{41}{256}a^{3}-\frac{5}{64}a^{2}-\frac{1}{8}a$, $\frac{1}{40960}a^{12}+\frac{19}{40960}a^{11}-\frac{15}{8192}a^{10}+\frac{107}{40960}a^{9}+\frac{107}{40960}a^{8}-\frac{531}{40960}a^{7}+\frac{43}{8192}a^{6}+\frac{857}{40960}a^{5}-\frac{161}{5120}a^{4}+\frac{1711}{10240}a^{3}+\frac{13}{512}a^{2}-\frac{121}{320}a-\frac{2}{5}$, $\frac{1}{163840}a^{13}-\frac{9}{40960}a^{11}-\frac{17}{40960}a^{10}-\frac{283}{81920}a^{9}-\frac{1}{40960}a^{8}+\frac{259}{10240}a^{7}+\frac{1033}{40960}a^{6}+\frac{6989}{163840}a^{5}-\frac{1931}{40960}a^{4}-\frac{2729}{40960}a^{3}-\frac{283}{10240}a^{2}-\frac{509}{1280}a-\frac{7}{20}$, $\frac{1}{45875200}a^{14}-\frac{13}{5734400}a^{13}-\frac{37}{11468800}a^{12}+\frac{507}{11468800}a^{11}+\frac{37213}{22937600}a^{10}-\frac{3383}{1638400}a^{9}-\frac{803}{358400}a^{8}+\frac{210117}{11468800}a^{7}+\frac{1206301}{45875200}a^{6}+\frac{709839}{11468800}a^{5}-\frac{536881}{11468800}a^{4}-\frac{415357}{2867200}a^{3}-\frac{297}{8960}a^{2}+\frac{1007}{8960}a-\frac{131}{700}$, $\frac{1}{18\!\cdots\!00}a^{15}-\frac{10\!\cdots\!81}{18\!\cdots\!00}a^{14}-\frac{62\!\cdots\!07}{93\!\cdots\!60}a^{13}-\frac{82\!\cdots\!09}{83\!\cdots\!00}a^{12}-\frac{64\!\cdots\!29}{18\!\cdots\!20}a^{11}-\frac{14\!\cdots\!63}{93\!\cdots\!00}a^{10}+\frac{89\!\cdots\!21}{46\!\cdots\!00}a^{9}-\frac{11\!\cdots\!11}{46\!\cdots\!00}a^{8}-\frac{11\!\cdots\!77}{53\!\cdots\!20}a^{7}-\frac{26\!\cdots\!21}{18\!\cdots\!00}a^{6}+\frac{37\!\cdots\!71}{26\!\cdots\!00}a^{5}+\frac{26\!\cdots\!89}{46\!\cdots\!00}a^{4}+\frac{21\!\cdots\!89}{11\!\cdots\!00}a^{3}-\frac{73\!\cdots\!93}{72\!\cdots\!20}a^{2}-\frac{74\!\cdots\!99}{18\!\cdots\!00}a-\frac{81\!\cdots\!23}{25\!\cdots\!00}$ Copy content Toggle raw display

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

$C_{301}\times C_{3913}$, which has order $1177813$ (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:   $\frac{55\!\cdots\!01}{28\!\cdots\!04}a^{15}-\frac{20\!\cdots\!69}{28\!\cdots\!04}a^{14}+\frac{69\!\cdots\!53}{70\!\cdots\!76}a^{13}-\frac{40\!\cdots\!79}{44\!\cdots\!36}a^{12}+\frac{10\!\cdots\!23}{14\!\cdots\!52}a^{11}-\frac{51\!\cdots\!39}{14\!\cdots\!52}a^{10}+\frac{75\!\cdots\!25}{70\!\cdots\!76}a^{9}-\frac{17\!\cdots\!11}{70\!\cdots\!76}a^{8}+\frac{36\!\cdots\!85}{28\!\cdots\!04}a^{7}-\frac{13\!\cdots\!01}{28\!\cdots\!04}a^{6}+\frac{44\!\cdots\!89}{80\!\cdots\!52}a^{5}-\frac{45\!\cdots\!15}{70\!\cdots\!76}a^{4}+\frac{18\!\cdots\!97}{17\!\cdots\!44}a^{3}-\frac{15\!\cdots\!65}{55\!\cdots\!92}a^{2}+\frac{80\!\cdots\!65}{27\!\cdots\!96}a-\frac{37\!\cdots\!47}{39\!\cdots\!24}$, $\frac{12\!\cdots\!11}{93\!\cdots\!00}a^{15}-\frac{21\!\cdots\!83}{18\!\cdots\!20}a^{14}+\frac{18\!\cdots\!37}{33\!\cdots\!00}a^{13}-\frac{29\!\cdots\!97}{29\!\cdots\!00}a^{12}+\frac{40\!\cdots\!67}{66\!\cdots\!00}a^{11}-\frac{73\!\cdots\!17}{18\!\cdots\!32}a^{10}+\frac{55\!\cdots\!03}{46\!\cdots\!80}a^{9}-\frac{53\!\cdots\!37}{23\!\cdots\!00}a^{8}+\frac{13\!\cdots\!23}{93\!\cdots\!00}a^{7}-\frac{11\!\cdots\!37}{26\!\cdots\!60}a^{6}+\frac{23\!\cdots\!97}{26\!\cdots\!80}a^{5}-\frac{26\!\cdots\!97}{23\!\cdots\!00}a^{4}-\frac{16\!\cdots\!53}{58\!\cdots\!00}a^{3}-\frac{21\!\cdots\!31}{36\!\cdots\!60}a^{2}-\frac{10\!\cdots\!09}{90\!\cdots\!00}a-\frac{34\!\cdots\!49}{12\!\cdots\!00}$, $\frac{96\!\cdots\!43}{93\!\cdots\!00}a^{15}-\frac{14\!\cdots\!79}{93\!\cdots\!00}a^{14}+\frac{11\!\cdots\!11}{23\!\cdots\!00}a^{13}-\frac{19\!\cdots\!89}{58\!\cdots\!60}a^{12}+\frac{11\!\cdots\!61}{46\!\cdots\!00}a^{11}-\frac{49\!\cdots\!77}{46\!\cdots\!00}a^{10}+\frac{29\!\cdots\!79}{23\!\cdots\!00}a^{9}-\frac{17\!\cdots\!17}{23\!\cdots\!00}a^{8}+\frac{37\!\cdots\!67}{93\!\cdots\!00}a^{7}-\frac{60\!\cdots\!39}{93\!\cdots\!00}a^{6}+\frac{13\!\cdots\!37}{18\!\cdots\!00}a^{5}-\frac{32\!\cdots\!51}{33\!\cdots\!00}a^{4}+\frac{23\!\cdots\!97}{83\!\cdots\!00}a^{3}-\frac{46\!\cdots\!59}{36\!\cdots\!60}a^{2}+\frac{11\!\cdots\!63}{90\!\cdots\!00}a-\frac{65\!\cdots\!53}{12\!\cdots\!00}$, $\frac{10\!\cdots\!87}{93\!\cdots\!60}a^{15}+\frac{85\!\cdots\!33}{46\!\cdots\!00}a^{14}+\frac{80\!\cdots\!87}{11\!\cdots\!00}a^{13}-\frac{33\!\cdots\!67}{14\!\cdots\!00}a^{12}+\frac{67\!\cdots\!77}{23\!\cdots\!00}a^{11}-\frac{20\!\cdots\!01}{23\!\cdots\!00}a^{10}+\frac{22\!\cdots\!87}{11\!\cdots\!00}a^{9}-\frac{17\!\cdots\!99}{16\!\cdots\!00}a^{8}+\frac{13\!\cdots\!99}{46\!\cdots\!00}a^{7}-\frac{46\!\cdots\!27}{46\!\cdots\!00}a^{6}+\frac{17\!\cdots\!43}{16\!\cdots\!00}a^{5}-\frac{20\!\cdots\!73}{11\!\cdots\!00}a^{4}+\frac{53\!\cdots\!39}{29\!\cdots\!00}a^{3}-\frac{15\!\cdots\!89}{25\!\cdots\!40}a^{2}+\frac{42\!\cdots\!19}{90\!\cdots\!40}a-\frac{17\!\cdots\!33}{64\!\cdots\!00}$, $\frac{54\!\cdots\!89}{93\!\cdots\!00}a^{15}-\frac{16\!\cdots\!93}{93\!\cdots\!00}a^{14}+\frac{10\!\cdots\!07}{33\!\cdots\!00}a^{13}-\frac{77\!\cdots\!11}{29\!\cdots\!00}a^{12}+\frac{13\!\cdots\!77}{66\!\cdots\!00}a^{11}-\frac{52\!\cdots\!79}{46\!\cdots\!00}a^{10}+\frac{78\!\cdots\!13}{23\!\cdots\!00}a^{9}-\frac{46\!\cdots\!63}{46\!\cdots\!80}a^{8}+\frac{34\!\cdots\!73}{93\!\cdots\!00}a^{7}-\frac{15\!\cdots\!59}{13\!\cdots\!00}a^{6}+\frac{36\!\cdots\!83}{13\!\cdots\!00}a^{5}-\frac{30\!\cdots\!11}{93\!\cdots\!16}a^{4}+\frac{60\!\cdots\!49}{58\!\cdots\!00}a^{3}+\frac{98\!\cdots\!83}{36\!\cdots\!60}a^{2}-\frac{12\!\cdots\!91}{90\!\cdots\!00}a+\frac{25\!\cdots\!77}{12\!\cdots\!00}$, $\frac{52\!\cdots\!29}{93\!\cdots\!00}a^{15}-\frac{20\!\cdots\!19}{13\!\cdots\!00}a^{14}+\frac{64\!\cdots\!49}{23\!\cdots\!00}a^{13}-\frac{49\!\cdots\!91}{29\!\cdots\!00}a^{12}+\frac{61\!\cdots\!99}{46\!\cdots\!00}a^{11}-\frac{26\!\cdots\!99}{46\!\cdots\!00}a^{10}+\frac{14\!\cdots\!93}{23\!\cdots\!00}a^{9}-\frac{15\!\cdots\!63}{46\!\cdots\!80}a^{8}+\frac{14\!\cdots\!93}{93\!\cdots\!00}a^{7}-\frac{33\!\cdots\!53}{93\!\cdots\!00}a^{6}+\frac{34\!\cdots\!13}{13\!\cdots\!00}a^{5}+\frac{16\!\cdots\!45}{93\!\cdots\!16}a^{4}-\frac{74\!\cdots\!71}{58\!\cdots\!00}a^{3}+\frac{20\!\cdots\!63}{36\!\cdots\!60}a^{2}-\frac{95\!\cdots\!73}{12\!\cdots\!00}a+\frac{21\!\cdots\!37}{12\!\cdots\!00}$, $\frac{14\!\cdots\!33}{93\!\cdots\!00}a^{15}-\frac{27\!\cdots\!11}{26\!\cdots\!60}a^{14}+\frac{20\!\cdots\!77}{23\!\cdots\!00}a^{13}-\frac{14\!\cdots\!91}{29\!\cdots\!00}a^{12}+\frac{21\!\cdots\!87}{46\!\cdots\!00}a^{11}-\frac{20\!\cdots\!31}{93\!\cdots\!60}a^{10}+\frac{27\!\cdots\!53}{46\!\cdots\!80}a^{9}-\frac{56\!\cdots\!71}{23\!\cdots\!00}a^{8}+\frac{61\!\cdots\!29}{93\!\cdots\!00}a^{7}-\frac{49\!\cdots\!29}{18\!\cdots\!20}a^{6}+\frac{37\!\cdots\!49}{52\!\cdots\!16}a^{5}-\frac{17\!\cdots\!51}{23\!\cdots\!00}a^{4}+\frac{34\!\cdots\!01}{58\!\cdots\!00}a^{3}+\frac{30\!\cdots\!03}{72\!\cdots\!72}a^{2}+\frac{44\!\cdots\!99}{12\!\cdots\!00}a+\frac{28\!\cdots\!13}{12\!\cdots\!00}$ 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:  \( 325756301.284 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 325756301.284 \cdot 1177813}{2\cdot\sqrt{1722945145525970304337872614249761}}\cr\approx \mathstrut & 11226.4486590 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856)
 
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 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856, 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 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856);
 
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 - 2*x^15 + 53*x^14 - 404*x^13 + 3150*x^12 - 16220*x^11 + 43602*x^10 - 147216*x^9 + 562581*x^8 - 1745730*x^7 + 3929497*x^6 - 4677356*x^5 + 6836220*x^4 - 10615248*x^3 + 18312704*x^2 - 3638272*x + 7929856);
 
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

$C_2^2:C_4$ (as 16T10):

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 16
The 10 conjugacy class representatives for $C_2^2 : C_4$
Character table for $C_2^2 : C_4$

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{89}) \), \(\Q(\sqrt{1513}) \), \(\Q(\sqrt{17}, \sqrt{89})\), 4.0.11984473.1 x2, 4.0.203736041.1 x2, 4.0.25721.1 x2, 4.0.134657.1 x2, 4.4.704969.1, 4.4.203736041.1, 8.0.41508374402353681.3, 8.0.5240294710561.2, 8.8.41508374402353681.1, 8.0.143627593087729.1 x2, 8.0.41508374402353681.1 x2

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 8 siblings: 8.0.143627593087729.1, 8.0.41508374402353681.1
Minimal sibling: 8.0.143627593087729.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.1.0.1}{1} }^{16}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ ${\href{/padicField/5.2.0.1}{2} }^{8}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.2.0.1}{2} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ R ${\href{/padicField/19.4.0.1}{4} }^{4}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.2.0.1}{2} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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
\(17\) Copy content Toggle raw display 17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(89\) Copy content Toggle raw display 89.8.6.1$x^{8} + 328 x^{7} + 40356 x^{6} + 2208424 x^{5} + 45454472 x^{4} + 6654464 x^{3} + 3950616 x^{2} + 196033136 x + 4016712036$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
89.8.6.1$x^{8} + 328 x^{7} + 40356 x^{6} + 2208424 x^{5} + 45454472 x^{4} + 6654464 x^{3} + 3950616 x^{2} + 196033136 x + 4016712036$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$