Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196)
gp: K = bnfinit(y^16 - 3*y^15 + 37*y^14 - 104*y^13 + 850*y^12 - 2306*y^11 + 12064*y^10 - 30644*y^9 + 122068*y^8 - 304134*y^7 + 826306*y^6 - 1935990*y^5 + 4270839*y^4 - 7629345*y^3 + 12479679*y^2 - 14077098*y + 16112196, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196)
\( x^{16} - 3 x^{15} + 37 x^{14} - 104 x^{13} + 850 x^{12} - 2306 x^{11} + 12064 x^{10} - 30644 x^{9} + \cdots + 16112196 \)
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: |
\(800737337114777368288115578449\)
\(\medspace = 3^{8}\cdot 73^{14}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(73.96\) | 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: | $3^{1/2}73^{7/8}\approx 73.95510022503493$ | ||
| Ramified primes: |
\(3\), \(73\)
| 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 not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{6}a^{6}+\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{3}a^{2}-\frac{1}{2}a$, $\frac{1}{6}a^{7}+\frac{1}{6}a^{5}-\frac{1}{3}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{36}a^{8}+\frac{1}{18}a^{7}-\frac{1}{18}a^{6}+\frac{2}{9}a^{5}-\frac{1}{18}a^{4}-\frac{5}{18}a^{3}+\frac{5}{12}a^{2}-\frac{1}{3}a$, $\frac{1}{36}a^{9}+\frac{1}{6}a^{5}-\frac{13}{36}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{36}a^{10}-\frac{1}{36}a^{4}$, $\frac{1}{144}a^{11}+\frac{1}{144}a^{8}-\frac{5}{72}a^{7}-\frac{1}{18}a^{6}-\frac{5}{144}a^{5}+\frac{7}{36}a^{4}-\frac{1}{36}a^{3}-\frac{5}{16}a^{2}-\frac{11}{24}a-\frac{1}{4}$, $\frac{1}{432}a^{12}+\frac{1}{108}a^{10}+\frac{5}{432}a^{9}-\frac{1}{216}a^{8}-\frac{1}{27}a^{7}-\frac{7}{144}a^{6}-\frac{13}{108}a^{5}-\frac{2}{9}a^{4}+\frac{5}{144}a^{3}+\frac{1}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{432}a^{13}+\frac{1}{432}a^{11}+\frac{5}{432}a^{10}-\frac{1}{216}a^{9}+\frac{5}{432}a^{8}-\frac{5}{144}a^{7}-\frac{1}{108}a^{6}+\frac{13}{144}a^{5}-\frac{5}{48}a^{4}+\frac{31}{72}a^{3}-\frac{7}{16}a^{2}-\frac{5}{24}a+\frac{1}{4}$, $\frac{1}{1093824}a^{14}-\frac{161}{546912}a^{13}+\frac{497}{546912}a^{12}+\frac{71}{34182}a^{11}+\frac{167}{273456}a^{10}-\frac{4657}{546912}a^{9}+\frac{395}{546912}a^{8}-\frac{2933}{136728}a^{7}+\frac{16841}{546912}a^{6}-\frac{11099}{45576}a^{5}+\frac{4397}{91152}a^{4}-\frac{18875}{60768}a^{3}-\frac{42071}{121536}a^{2}-\frac{1157}{20256}a-\frac{323}{3376}$, $\frac{1}{19\!\cdots\!68}a^{15}+\frac{29\!\cdots\!57}{95\!\cdots\!84}a^{14}-\frac{60\!\cdots\!85}{95\!\cdots\!84}a^{13}-\frac{60\!\cdots\!57}{14\!\cdots\!91}a^{12}-\frac{29\!\cdots\!07}{23\!\cdots\!96}a^{11}-\frac{10\!\cdots\!31}{95\!\cdots\!84}a^{10}-\frac{41\!\cdots\!55}{31\!\cdots\!28}a^{9}-\frac{18\!\cdots\!77}{47\!\cdots\!92}a^{8}+\frac{28\!\cdots\!59}{95\!\cdots\!84}a^{7}+\frac{18\!\cdots\!39}{23\!\cdots\!96}a^{6}+\frac{40\!\cdots\!43}{39\!\cdots\!16}a^{5}+\frac{11\!\cdots\!41}{31\!\cdots\!28}a^{4}+\frac{44\!\cdots\!11}{70\!\cdots\!84}a^{3}-\frac{27\!\cdots\!23}{10\!\cdots\!76}a^{2}+\frac{34\!\cdots\!95}{17\!\cdots\!96}a+\frac{55\!\cdots\!68}{16\!\cdots\!99}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
$C_{4}$, which has order $4$ (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: |
\( \frac{14499826137671475455}{226030182627177499055524272} a^{15} - \frac{15987868879075745887}{56507545656794374763881068} a^{14} + \frac{680348569009555995127}{226030182627177499055524272} a^{13} - \frac{1286431468471518157}{121325916600739398312144} a^{12} + \frac{16660316977632515312825}{226030182627177499055524272} a^{11} - \frac{53291839986934039605113}{226030182627177499055524272} a^{10} + \frac{88147140135779286742243}{75343394209059166351841424} a^{9} - \frac{734145426492639988891597}{226030182627177499055524272} a^{8} + \frac{2795745413489208422453131}{226030182627177499055524272} a^{7} - \frac{7023986935344381549435293}{226030182627177499055524272} a^{6} + \frac{6919529768917670372069153}{75343394209059166351841424} a^{5} - \frac{15168196515807677637694225}{75343394209059166351841424} a^{4} + \frac{932710014502687642912481}{2092872061362754620884484} a^{3} - \frac{17501585202279771235073819}{25114464736353055450613808} a^{2} + \frac{5497288394930086996098583}{4185744122725509241768968} a - \frac{1614075467195279358845}{3128358836117719911636} \)
(order $6$)
| 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: |
$\frac{20\!\cdots\!57}{15\!\cdots\!64}a^{15}-\frac{25\!\cdots\!99}{15\!\cdots\!64}a^{14}+\frac{70\!\cdots\!29}{79\!\cdots\!32}a^{13}-\frac{53\!\cdots\!29}{11\!\cdots\!28}a^{12}+\frac{14\!\cdots\!97}{79\!\cdots\!32}a^{11}-\frac{17\!\cdots\!65}{19\!\cdots\!58}a^{10}+\frac{19\!\cdots\!21}{66\!\cdots\!86}a^{9}-\frac{21\!\cdots\!77}{19\!\cdots\!58}a^{8}+\frac{57\!\cdots\!55}{19\!\cdots\!58}a^{7}-\frac{73\!\cdots\!65}{79\!\cdots\!32}a^{6}+\frac{69\!\cdots\!13}{26\!\cdots\!44}a^{5}-\frac{21\!\cdots\!53}{33\!\cdots\!93}a^{4}+\frac{73\!\cdots\!73}{58\!\cdots\!32}a^{3}-\frac{36\!\cdots\!77}{17\!\cdots\!96}a^{2}+\frac{72\!\cdots\!17}{29\!\cdots\!16}a-\frac{45\!\cdots\!77}{22\!\cdots\!32}$, $\frac{82\!\cdots\!99}{42\!\cdots\!08}a^{15}-\frac{48\!\cdots\!83}{42\!\cdots\!08}a^{14}+\frac{14\!\cdots\!13}{21\!\cdots\!04}a^{13}-\frac{88\!\cdots\!95}{25\!\cdots\!68}a^{12}+\frac{32\!\cdots\!93}{21\!\cdots\!04}a^{11}-\frac{38\!\cdots\!47}{53\!\cdots\!76}a^{10}+\frac{14\!\cdots\!07}{71\!\cdots\!68}a^{9}-\frac{88\!\cdots\!25}{10\!\cdots\!52}a^{8}+\frac{10\!\cdots\!35}{53\!\cdots\!76}a^{7}-\frac{77\!\cdots\!53}{10\!\cdots\!52}a^{6}+\frac{10\!\cdots\!97}{71\!\cdots\!68}a^{5}-\frac{14\!\cdots\!23}{44\!\cdots\!73}a^{4}+\frac{12\!\cdots\!77}{15\!\cdots\!04}a^{3}-\frac{47\!\cdots\!65}{47\!\cdots\!12}a^{2}+\frac{46\!\cdots\!63}{37\!\cdots\!32}a-\frac{11\!\cdots\!67}{13\!\cdots\!92}$, $\frac{84\!\cdots\!77}{82\!\cdots\!16}a^{15}-\frac{23\!\cdots\!41}{41\!\cdots\!08}a^{14}+\frac{19\!\cdots\!77}{41\!\cdots\!08}a^{13}-\frac{46\!\cdots\!33}{23\!\cdots\!56}a^{12}+\frac{23\!\cdots\!77}{20\!\cdots\!04}a^{11}-\frac{18\!\cdots\!73}{41\!\cdots\!08}a^{10}+\frac{24\!\cdots\!31}{13\!\cdots\!36}a^{9}-\frac{61\!\cdots\!89}{10\!\cdots\!52}a^{8}+\frac{79\!\cdots\!69}{41\!\cdots\!08}a^{7}-\frac{11\!\cdots\!33}{20\!\cdots\!04}a^{6}+\frac{96\!\cdots\!67}{69\!\cdots\!68}a^{5}-\frac{44\!\cdots\!45}{13\!\cdots\!36}a^{4}+\frac{21\!\cdots\!39}{30\!\cdots\!08}a^{3}-\frac{59\!\cdots\!31}{46\!\cdots\!12}a^{2}+\frac{12\!\cdots\!47}{76\!\cdots\!52}a-\frac{14\!\cdots\!35}{14\!\cdots\!26}$, $\frac{12\!\cdots\!01}{19\!\cdots\!68}a^{15}-\frac{13\!\cdots\!21}{19\!\cdots\!68}a^{14}+\frac{96\!\cdots\!49}{23\!\cdots\!96}a^{13}-\frac{12\!\cdots\!01}{51\!\cdots\!68}a^{12}+\frac{12\!\cdots\!95}{11\!\cdots\!48}a^{11}-\frac{49\!\cdots\!71}{95\!\cdots\!84}a^{10}+\frac{28\!\cdots\!71}{15\!\cdots\!64}a^{9}-\frac{63\!\cdots\!91}{95\!\cdots\!84}a^{8}+\frac{18\!\cdots\!85}{95\!\cdots\!84}a^{7}-\frac{59\!\cdots\!23}{95\!\cdots\!84}a^{6}+\frac{62\!\cdots\!31}{39\!\cdots\!16}a^{5}-\frac{11\!\cdots\!31}{31\!\cdots\!28}a^{4}+\frac{54\!\cdots\!05}{70\!\cdots\!84}a^{3}-\frac{32\!\cdots\!01}{21\!\cdots\!52}a^{2}+\frac{78\!\cdots\!77}{35\!\cdots\!92}a-\frac{51\!\cdots\!73}{26\!\cdots\!84}$, $\frac{86\!\cdots\!33}{47\!\cdots\!92}a^{15}-\frac{64\!\cdots\!71}{95\!\cdots\!84}a^{14}+\frac{14\!\cdots\!65}{23\!\cdots\!96}a^{13}-\frac{28\!\cdots\!27}{11\!\cdots\!28}a^{12}+\frac{66\!\cdots\!45}{47\!\cdots\!92}a^{11}-\frac{25\!\cdots\!23}{47\!\cdots\!92}a^{10}+\frac{36\!\cdots\!53}{19\!\cdots\!58}a^{9}-\frac{15\!\cdots\!47}{23\!\cdots\!96}a^{8}+\frac{85\!\cdots\!47}{47\!\cdots\!92}a^{7}-\frac{99\!\cdots\!97}{11\!\cdots\!48}a^{6}+\frac{16\!\cdots\!73}{15\!\cdots\!64}a^{5}-\frac{14\!\cdots\!47}{15\!\cdots\!64}a^{4}+\frac{67\!\cdots\!05}{17\!\cdots\!96}a^{3}-\frac{39\!\cdots\!41}{10\!\cdots\!76}a^{2}+\frac{12\!\cdots\!85}{17\!\cdots\!96}a-\frac{92\!\cdots\!63}{13\!\cdots\!92}$, $\frac{50\!\cdots\!69}{21\!\cdots\!52}a^{15}+\frac{18\!\cdots\!73}{21\!\cdots\!52}a^{14}+\frac{10\!\cdots\!21}{13\!\cdots\!72}a^{13}+\frac{27\!\cdots\!19}{15\!\cdots\!04}a^{12}+\frac{33\!\cdots\!63}{26\!\cdots\!44}a^{11}+\frac{18\!\cdots\!27}{10\!\cdots\!76}a^{10}+\frac{85\!\cdots\!69}{98\!\cdots\!72}a^{9}-\frac{11\!\cdots\!97}{10\!\cdots\!76}a^{8}+\frac{25\!\cdots\!03}{10\!\cdots\!76}a^{7}-\frac{28\!\cdots\!69}{10\!\cdots\!76}a^{6}+\frac{82\!\cdots\!07}{88\!\cdots\!48}a^{5}+\frac{17\!\cdots\!63}{35\!\cdots\!92}a^{4}+\frac{11\!\cdots\!39}{23\!\cdots\!28}a^{3}+\frac{33\!\cdots\!77}{23\!\cdots\!28}a^{2}-\frac{11\!\cdots\!45}{39\!\cdots\!88}a+\frac{16\!\cdots\!13}{29\!\cdots\!76}$, $\frac{63\!\cdots\!91}{19\!\cdots\!68}a^{15}+\frac{51\!\cdots\!11}{95\!\cdots\!84}a^{14}+\frac{98\!\cdots\!97}{95\!\cdots\!84}a^{13}+\frac{48\!\cdots\!49}{11\!\cdots\!28}a^{12}+\frac{78\!\cdots\!45}{47\!\cdots\!92}a^{11}-\frac{96\!\cdots\!29}{95\!\cdots\!84}a^{10}+\frac{44\!\cdots\!75}{31\!\cdots\!28}a^{9}-\frac{36\!\cdots\!67}{11\!\cdots\!48}a^{8}+\frac{89\!\cdots\!69}{95\!\cdots\!84}a^{7}-\frac{27\!\cdots\!57}{11\!\cdots\!48}a^{6}+\frac{95\!\cdots\!43}{15\!\cdots\!64}a^{5}-\frac{36\!\cdots\!69}{31\!\cdots\!28}a^{4}+\frac{16\!\cdots\!33}{70\!\cdots\!84}a^{3}-\frac{28\!\cdots\!11}{10\!\cdots\!76}a^{2}+\frac{86\!\cdots\!03}{17\!\cdots\!96}a-\frac{71\!\cdots\!93}{66\!\cdots\!96}$
| 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: | \( 20892766922.9 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 20892766922.9 \cdot 4}{6\cdot\sqrt{800737337114777368288115578449}}\cr\approx \mathstrut & 37.8092974334 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196)
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 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196, 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 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196);
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 - 3*x^15 + 37*x^14 - 104*x^13 + 850*x^12 - 2306*x^11 + 12064*x^10 - 30644*x^9 + 122068*x^8 - 304134*x^7 + 826306*x^6 - 1935990*x^5 + 4270839*x^4 - 7629345*x^3 + 12479679*x^2 - 14077098*x + 16112196);
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
$\SD_{16}$ (as 16T12):
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 7 conjugacy class representatives for $QD_{16}$ |
| Character table for $QD_{16}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{-219}) \), \(\Q(\sqrt{-3}, \sqrt{73})\), 4.2.1167051.1 x2, 4.0.3501153.2 x2, 8.0.12258072329409.3, 8.2.298279760015619.1 x4 |
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
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.2.0.1}{2} }^{8}$ | R | ${\href{/padicField/5.4.0.1}{4} }^{4}$ | ${\href{/padicField/7.8.0.1}{8} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}$ | ${\href{/padicField/23.2.0.1}{2} }^{8}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.8.0.1}{8} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.8.0.1}{8} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(73\)
| 73.8.7.4 | $x^{8} + 1825$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
| 73.8.7.4 | $x^{8} + 1825$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |