Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1)
gp: K = bnfinit(y^23 - 6*y^22 - 5*y^21 + 86*y^20 - 45*y^19 - 531*y^18 + 490*y^17 + 1869*y^16 - 1929*y^15 - 4166*y^14 + 4062*y^13 + 6124*y^12 - 4949*y^11 - 5930*y^10 + 3501*y^9 + 3657*y^8 - 1386*y^7 - 1348*y^6 + 286*y^5 + 273*y^4 - 28*y^3 - 27*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1)
\( x^{23} - 6 x^{22} - 5 x^{21} + 86 x^{20} - 45 x^{19} - 531 x^{18} + 490 x^{17} + 1869 x^{16} - 1929 x^{15} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[15, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(9024061084396593245740482274651471479637\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(54.60\) | 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: | $9024061084396593245740482274651471479637^{1/2}\approx 9.499505821039636e+19$ | ||
| Ramified primes: |
\(90240\!\cdots\!79637\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{90240\!\cdots\!79637}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{53}a^{22}+\frac{2}{53}a^{21}+\frac{11}{53}a^{20}+\frac{15}{53}a^{19}+\frac{22}{53}a^{18}+\frac{16}{53}a^{17}-\frac{18}{53}a^{16}-\frac{24}{53}a^{15}-\frac{1}{53}a^{14}+\frac{13}{53}a^{13}-\frac{21}{53}a^{12}+\frac{20}{53}a^{11}-\frac{19}{53}a^{10}+\frac{13}{53}a^{9}+\frac{1}{53}a^{8}+\frac{8}{53}a^{7}+\frac{3}{53}a^{6}+\frac{1}{53}a^{5}-\frac{24}{53}a^{4}-\frac{25}{53}a^{3}-\frac{16}{53}a^{2}+\frac{4}{53}a-\frac{20}{53}$
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
Trivial group, which has order $1$ (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: | $18$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$a$, $a-1$, $a^{22}-6a^{21}-5a^{20}+86a^{19}-45a^{18}-531a^{17}+490a^{16}+1869a^{15}-1929a^{14}-4166a^{13}+4062a^{12}+6124a^{11}-4949a^{10}-5930a^{9}+3501a^{8}+3657a^{7}-1386a^{6}-1348a^{5}+286a^{4}+272a^{3}-27a^{2}-24a$, $a^{22}-6a^{21}-5a^{20}+86a^{19}-45a^{18}-531a^{17}+490a^{16}+1869a^{15}-1929a^{14}-4166a^{13}+4062a^{12}+6124a^{11}-4949a^{10}-5930a^{9}+3501a^{8}+3657a^{7}-1386a^{6}-1348a^{5}+286a^{4}+272a^{3}-27a^{2}-24a-1$, $\frac{1583}{53}a^{22}-\frac{10137}{53}a^{21}-\frac{3946}{53}a^{20}+\frac{138384}{53}a^{19}-\frac{126082}{53}a^{18}-\frac{799299}{53}a^{17}+\frac{1097703}{53}a^{16}+\frac{2576551}{53}a^{15}-\frac{4112952}{53}a^{14}-\frac{5152804}{53}a^{13}+\frac{8593832}{53}a^{12}+\frac{6704996}{53}a^{11}-\frac{10691133}{53}a^{10}-\frac{5727907}{53}a^{9}+\frac{7980415}{53}a^{8}+\frac{3109242}{53}a^{7}-\frac{3494417}{53}a^{6}-\frac{979871}{53}a^{5}+\frac{854157}{53}a^{4}+\frac{154140}{53}a^{3}-\frac{107001}{53}a^{2}-\frac{8561}{53}a+\frac{5122}{53}$, $\frac{340}{53}a^{22}-\frac{2235}{53}a^{21}-\frac{659}{53}a^{20}+\frac{30805}{53}a^{19}-\frac{30521}{53}a^{18}-\frac{180961}{53}a^{17}+\frac{262219}{53}a^{16}+\frac{601764}{53}a^{15}-\frac{993401}{53}a^{14}-\frac{1273145}{53}a^{13}+\frac{2109097}{53}a^{12}+\frac{1817174}{53}a^{11}-\frac{2650153}{53}a^{10}-\frac{1762812}{53}a^{9}+\frac{1951323}{53}a^{8}+\frac{1095527}{53}a^{7}-\frac{804845}{53}a^{6}-\frac{384016}{53}a^{5}+\frac{175644}{53}a^{4}+\frac{63315}{53}a^{3}-\frac{19379}{53}a^{2}-\frac{3463}{53}a+\frac{832}{53}$, $\frac{195}{53}a^{22}-\frac{935}{53}a^{21}-\frac{2360}{53}a^{20}+\frac{15486}{53}a^{19}+\frac{11074}{53}a^{18}-\frac{112208}{53}a^{17}-\frac{27042}{53}a^{16}+\frac{465218}{53}a^{15}+\frac{55190}{53}a^{14}-\frac{1205123}{53}a^{13}-\frac{165586}{53}a^{12}+\frac{1993202}{53}a^{11}+\frac{424323}{53}a^{10}-\frac{2070136}{53}a^{9}-\frac{621972}{53}a^{8}+\frac{1289990}{53}a^{7}+\frac{487602}{53}a^{6}-\frac{444369}{53}a^{5}-\frac{194102}{53}a^{4}+\frac{73300}{53}a^{3}+\frac{34987}{53}a^{2}-\frac{4308}{53}a-\frac{2151}{53}$, $\frac{1335}{53}a^{22}-\frac{8513}{53}a^{21}-\frac{3494}{53}a^{20}+\frac{116379}{53}a^{19}-\frac{104243}{53}a^{18}-\frac{672887}{53}a^{17}+\frac{915766}{53}a^{16}+\frac{2167990}{53}a^{15}-\frac{3450416}{53}a^{14}-\frac{4317886}{53}a^{13}+\frac{7267945}{53}a^{12}+\frac{5560324}{53}a^{11}-\frac{9181168}{53}a^{10}-\frac{4670124}{53}a^{9}+\frac{7060352}{53}a^{8}+\frac{2497387}{53}a^{7}-\frac{3256396}{53}a^{6}-\frac{791863}{53}a^{5}+\frac{856982}{53}a^{4}+\frac{129441}{53}a^{3}-\frac{115329}{53}a^{2}-\frac{7327}{53}a+\frac{5577}{53}$, $a^{3}-2a^{2}-a+1$, $\frac{2178}{53}a^{22}-\frac{13505}{53}a^{21}-\frac{7948}{53}a^{20}+\frac{187642}{53}a^{19}-\frac{137478}{53}a^{18}-\frac{1110323}{53}a^{17}+\frac{1289559}{53}a^{16}+\frac{3693238}{53}a^{15}-\frac{4894449}{53}a^{14}-\frac{7662039}{53}a^{13}+\frac{10180559}{53}a^{12}+\frac{10330754}{53}a^{11}-\frac{12461137}{53}a^{10}-\frac{9043749}{53}a^{9}+\frac{9058818}{53}a^{8}+\frac{4941124}{53}a^{7}-\frac{3825790}{53}a^{6}-\frac{1541447}{53}a^{5}+\frac{892930}{53}a^{4}+\frac{235354}{53}a^{3}-\frac{106133}{53}a^{2}-\frac{12382}{53}a+\frac{4776}{53}$, $\frac{79}{53}a^{22}-\frac{478}{53}a^{21}-\frac{244}{53}a^{20}+\frac{6008}{53}a^{19}-\frac{4198}{53}a^{18}-\frac{31066}{53}a^{17}+\frac{31067}{53}a^{16}+\frac{86190}{53}a^{15}-\frac{78466}{53}a^{14}-\frac{139688}{53}a^{13}+\frac{48797}{53}a^{12}+\frac{131483}{53}a^{11}+\frac{143984}{53}a^{10}-\frac{58863}{53}a^{9}-\frac{324175}{53}a^{8}-\frac{6364}{53}a^{7}+\frac{272498}{53}a^{6}+\frac{19053}{53}a^{5}-\frac{105882}{53}a^{4}-\frac{6374}{53}a^{3}+\frac{16650}{53}a^{2}+\frac{687}{53}a-\frac{732}{53}$, $\frac{647}{53}a^{22}-\frac{4271}{53}a^{21}-\frac{886}{53}a^{20}+\frac{57458}{53}a^{19}-\frac{62457}{53}a^{18}-\frac{324290}{53}a^{17}+\frac{519785}{53}a^{16}+\frac{1008644}{53}a^{15}-\frac{1945164}{53}a^{14}-\frac{1914217}{53}a^{13}+\frac{4128098}{53}a^{12}+\frac{2324058}{53}a^{11}-\frac{5295916}{53}a^{10}-\frac{1837155}{53}a^{9}+\frac{4146307}{53}a^{8}+\frac{931669}{53}a^{7}-\frac{1935156}{53}a^{6}-\frac{280465}{53}a^{5}+\frac{505833}{53}a^{4}+\frac{42602}{53}a^{3}-\frac{65525}{53}a^{2}-\frac{2235}{53}a+\frac{3066}{53}$, $\frac{50}{53}a^{22}-\frac{483}{53}a^{21}+\frac{868}{53}a^{20}+\frac{5096}{53}a^{19}-\frac{18086}{53}a^{18}-\frac{16637}{53}a^{17}+\frac{120735}{53}a^{16}-\frac{6182}{53}a^{15}-\frac{428396}{53}a^{14}+\frac{176822}{53}a^{13}+\frac{925761}{53}a^{12}-\frac{496458}{53}a^{11}-\frac{1283232}{53}a^{10}+\frac{666913}{53}a^{9}+\frac{1149832}{53}a^{8}-\frac{474215}{53}a^{7}-\frac{643853}{53}a^{6}+\frac{169703}{53}a^{5}+\frac{205871}{53}a^{4}-\frac{25471}{53}a^{3}-\frac{31752}{53}a^{2}+\frac{1048}{53}a+\frac{1650}{53}$, $\frac{122}{53}a^{22}-\frac{1081}{53}a^{21}+\frac{1395}{53}a^{20}+\frac{12642}{53}a^{19}-\frac{34416}{53}a^{18}-\frac{55447}{53}a^{17}+\frac{239378}{53}a^{16}+\frac{100369}{53}a^{15}-\frac{867732}{53}a^{14}-\frac{958}{53}a^{13}+\frac{1887789}{53}a^{12}-\frac{285562}{53}a^{11}-\frac{2579178}{53}a^{10}+\frac{460884}{53}a^{9}+\frac{2216264}{53}a^{8}-\frac{327942}{53}a^{7}-\frac{1169132}{53}a^{6}+\frac{112164}{53}a^{5}+\frac{357525}{53}a^{4}-\frac{13544}{53}a^{3}-\frac{54793}{53}a^{2}-\frac{42}{53}a+\frac{2913}{53}$, $\frac{891}{53}a^{22}-\frac{5267}{53}a^{21}-\frac{4933}{53}a^{20}+\frac{76382}{53}a^{19}-\frac{34087}{53}a^{18}-\frac{477319}{53}a^{17}+\frac{405524}{53}a^{16}+\frac{1696346}{53}a^{15}-\frac{1632549}{53}a^{14}-\frac{3790372}{53}a^{13}+\frac{3479554}{53}a^{12}+\frac{5505281}{53}a^{11}-\frac{4278076}{53}a^{10}-\frac{5139646}{53}a^{9}+\frac{3060528}{53}a^{8}+\frac{2934212}{53}a^{7}-\frac{1241290}{53}a^{6}-\frac{928570}{53}a^{5}+\frac{273879}{53}a^{4}+\frac{137308}{53}a^{3}-\frac{31269}{53}a^{2}-\frac{7195}{53}a+\frac{1472}{53}$, $\frac{345}{53}a^{22}-\frac{2013}{53}a^{21}-\frac{1982}{53}a^{20}+\frac{28972}{53}a^{19}-\frac{11543}{53}a^{18}-\frac{179344}{53}a^{17}+\frac{142296}{53}a^{16}+\frac{629628}{53}a^{15}-\frac{564212}{53}a^{14}-\frac{1384062}{53}a^{13}+\frac{1168719}{53}a^{12}+\frac{1967529}{53}a^{11}-\frac{1384184}{53}a^{10}-\frac{1792957}{53}a^{9}+\frac{950423}{53}a^{8}+\frac{1009654}{53}a^{7}-\frac{369647}{53}a^{6}-\frac{327354}{53}a^{5}+\frac{76626}{53}a^{4}+\frac{52802}{53}a^{3}-\frac{8170}{53}a^{2}-\frac{3019}{53}a+\frac{414}{53}$, $\frac{2867}{53}a^{22}-\frac{17904}{53}a^{21}-\frac{9697}{53}a^{20}+\frac{247585}{53}a^{19}-\frac{191750}{53}a^{18}-\frac{1455406}{53}a^{17}+\frac{1762584}{53}a^{16}+\frac{4798924}{53}a^{15}-\frac{6665497}{53}a^{14}-\frac{9849985}{53}a^{13}+\frac{13881602}{53}a^{12}+\frac{13130744}{53}a^{11}-\frac{17084433}{53}a^{10}-\frac{11387674}{53}a^{9}+\frac{12557719}{53}a^{8}+\frac{6201146}{53}a^{7}-\frac{5405985}{53}a^{6}-\frac{1951932}{53}a^{5}+\frac{1300447}{53}a^{4}+\frac{308335}{53}a^{3}-\frac{161306}{53}a^{2}-\frac{17576}{53}a+\frac{7744}{53}$, $\frac{1444}{53}a^{22}-\frac{10044}{53}a^{21}+\frac{1733}{53}a^{20}+\frac{126600}{53}a^{19}-\frac{184101}{53}a^{18}-\frac{646180}{53}a^{17}+\frac{1371406}{53}a^{16}+\frac{1706553}{53}a^{15}-\frac{4821317}{53}a^{14}-\frac{2429722}{53}a^{13}+\frac{9667881}{53}a^{12}+\frac{1636529}{53}a^{11}-\frac{11666342}{53}a^{10}-\frac{36560}{53}a^{9}+\frac{8501107}{53}a^{8}-\frac{643369}{53}a^{7}-\frac{3629666}{53}a^{6}+\frac{372073}{53}a^{5}+\frac{842229}{53}a^{4}-\frac{73465}{53}a^{3}-\frac{94442}{53}a^{2}+\frac{4186}{53}a+\frac{3821}{53}$
| 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: | \( 1818525117070 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{15}\cdot(2\pi)^{4}\cdot 1818525117070 \cdot 1}{2\cdot\sqrt{9024061084396593245740482274651471479637}}\cr\approx \mathstrut & 0.488829833620079 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1)
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^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1, 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^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1);
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^23 - 6*x^22 - 5*x^21 + 86*x^20 - 45*x^19 - 531*x^18 + 490*x^17 + 1869*x^16 - 1929*x^15 - 4166*x^14 + 4062*x^13 + 6124*x^12 - 4949*x^11 - 5930*x^10 + 3501*x^9 + 3657*x^8 - 1386*x^7 - 1348*x^6 + 286*x^5 + 273*x^4 - 28*x^3 - 27*x^2 + x + 1);
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
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 non-solvable group of order 25852016738884976640000 |
| The 1255 conjugacy class representatives for $S_{23}$ |
| Character table for $S_{23}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 46 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/7.6.0.1}{6} }$ | $18{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $23$ | $22{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.9.0.1}{9} }$ | $23$ | $23$ | ${\href{/padicField/41.14.0.1}{14} }{,}\,{\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $17{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(902\!\cdots\!637\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ |