Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 2*x^14 - 199*x^13 + 1245*x^12 + 5773*x^11 - 51145*x^10 + 64756*x^9 + 386511*x^8 - 2205812*x^7 + 4179448*x^6 + 12113345*x^5 - 63532217*x^4 + 35405533*x^3 + 195854391*x^2 - 355933406*x + 1368484)
gp: K = bnfinit(y^16 - 6*y^15 + 2*y^14 - 199*y^13 + 1245*y^12 + 5773*y^11 - 51145*y^10 + 64756*y^9 + 386511*y^8 - 2205812*y^7 + 4179448*y^6 + 12113345*y^5 - 63532217*y^4 + 35405533*y^3 + 195854391*y^2 - 355933406*y + 1368484, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 6*x^15 + 2*x^14 - 199*x^13 + 1245*x^12 + 5773*x^11 - 51145*x^10 + 64756*x^9 + 386511*x^8 - 2205812*x^7 + 4179448*x^6 + 12113345*x^5 - 63532217*x^4 + 35405533*x^3 + 195854391*x^2 - 355933406*x + 1368484);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 6*x^15 + 2*x^14 - 199*x^13 + 1245*x^12 + 5773*x^11 - 51145*x^10 + 64756*x^9 + 386511*x^8 - 2205812*x^7 + 4179448*x^6 + 12113345*x^5 - 63532217*x^4 + 35405533*x^3 + 195854391*x^2 - 355933406*x + 1368484)
\( x^{16} - 6 x^{15} + 2 x^{14} - 199 x^{13} + 1245 x^{12} + 5773 x^{11} - 51145 x^{10} + 64756 x^{9} + \cdots + 1368484 \)
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: | $[4, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1162337480711184271576898233831313\)
\(\medspace = 17^{15}\cdot 67^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(116.57\) | 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: | $17^{15/16}67^{1/2}\approx 116.56905215178652$ | ||
| Ramified primes: |
\(17\), \(67\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{10}-\frac{1}{4}a^{9}+\frac{1}{8}a^{8}+\frac{3}{8}a^{7}-\frac{3}{8}a^{6}-\frac{1}{2}a^{5}+\frac{1}{8}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{10}+\frac{3}{16}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{5}{16}a^{6}+\frac{1}{16}a^{4}+\frac{3}{8}a^{3}+\frac{3}{16}a^{2}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{64}a^{12}-\frac{1}{32}a^{11}-\frac{1}{16}a^{10}+\frac{9}{64}a^{9}+\frac{1}{8}a^{8}-\frac{31}{64}a^{7}-\frac{13}{64}a^{6}-\frac{15}{64}a^{5}-\frac{11}{64}a^{4}-\frac{27}{64}a^{3}+\frac{15}{64}a^{2}+\frac{1}{32}a+\frac{1}{16}$, $\frac{1}{256}a^{13}+\frac{1}{256}a^{12}-\frac{1}{128}a^{11}+\frac{5}{256}a^{10}+\frac{59}{256}a^{9}+\frac{41}{256}a^{8}-\frac{29}{128}a^{7}+\frac{17}{128}a^{6}-\frac{3}{32}a^{5}-\frac{13}{64}a^{4}+\frac{31}{128}a^{3}+\frac{119}{256}a^{2}+\frac{13}{128}a+\frac{11}{64}$, $\frac{1}{1024}a^{14}-\frac{3}{1024}a^{12}-\frac{25}{1024}a^{11}+\frac{11}{512}a^{10}+\frac{7}{512}a^{9}-\frac{35}{1024}a^{8}-\frac{57}{256}a^{7}-\frac{13}{512}a^{6}-\frac{71}{256}a^{5}-\frac{215}{512}a^{4}-\frac{455}{1024}a^{3}+\frac{259}{1024}a^{2}-\frac{87}{512}a+\frac{85}{256}$, $\frac{1}{67\!\cdots\!24}a^{15}-\frac{19\!\cdots\!33}{67\!\cdots\!24}a^{14}-\frac{17\!\cdots\!19}{67\!\cdots\!24}a^{13}-\frac{24\!\cdots\!03}{33\!\cdots\!12}a^{12}+\frac{16\!\cdots\!83}{67\!\cdots\!24}a^{11}+\frac{39\!\cdots\!71}{84\!\cdots\!28}a^{10}+\frac{11\!\cdots\!99}{67\!\cdots\!24}a^{9}-\frac{65\!\cdots\!09}{67\!\cdots\!24}a^{8}+\frac{12\!\cdots\!45}{33\!\cdots\!12}a^{7}-\frac{19\!\cdots\!77}{33\!\cdots\!12}a^{6}-\frac{10\!\cdots\!41}{33\!\cdots\!12}a^{5}+\frac{23\!\cdots\!07}{67\!\cdots\!24}a^{4}-\frac{10\!\cdots\!07}{33\!\cdots\!12}a^{3}-\frac{94\!\cdots\!13}{67\!\cdots\!24}a^{2}-\frac{11\!\cdots\!59}{33\!\cdots\!12}a-\frac{18\!\cdots\!97}{16\!\cdots\!56}$
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
$C_{2}\times C_{2}\times C_{4}\times C_{4}$, which has order $64$ (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: | $9$ | 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: |
$\frac{34\!\cdots\!01}{30\!\cdots\!68}a^{15}-\frac{34\!\cdots\!97}{30\!\cdots\!68}a^{14}+\frac{12\!\cdots\!09}{30\!\cdots\!68}a^{13}-\frac{51\!\cdots\!83}{15\!\cdots\!84}a^{12}+\frac{84\!\cdots\!15}{30\!\cdots\!68}a^{11}-\frac{10\!\cdots\!61}{37\!\cdots\!96}a^{10}-\frac{18\!\cdots\!85}{30\!\cdots\!68}a^{9}+\frac{82\!\cdots\!47}{30\!\cdots\!68}a^{8}-\frac{45\!\cdots\!35}{15\!\cdots\!84}a^{7}-\frac{24\!\cdots\!13}{15\!\cdots\!84}a^{6}+\frac{14\!\cdots\!59}{15\!\cdots\!84}a^{5}-\frac{44\!\cdots\!73}{30\!\cdots\!68}a^{4}-\frac{58\!\cdots\!83}{15\!\cdots\!84}a^{3}+\frac{32\!\cdots\!35}{30\!\cdots\!68}a^{2}-\frac{15\!\cdots\!31}{15\!\cdots\!84}a-\frac{36\!\cdots\!57}{75\!\cdots\!92}$, $\frac{52\!\cdots\!89}{15\!\cdots\!84}a^{15}-\frac{50\!\cdots\!01}{15\!\cdots\!84}a^{14}+\frac{16\!\cdots\!17}{15\!\cdots\!84}a^{13}-\frac{78\!\cdots\!35}{75\!\cdots\!92}a^{12}+\frac{12\!\cdots\!11}{15\!\cdots\!84}a^{11}-\frac{91\!\cdots\!69}{18\!\cdots\!48}a^{10}-\frac{25\!\cdots\!13}{15\!\cdots\!84}a^{9}+\frac{11\!\cdots\!79}{15\!\cdots\!84}a^{8}-\frac{68\!\cdots\!39}{75\!\cdots\!92}a^{7}-\frac{32\!\cdots\!49}{75\!\cdots\!92}a^{6}+\frac{20\!\cdots\!59}{75\!\cdots\!92}a^{5}-\frac{58\!\cdots\!85}{15\!\cdots\!84}a^{4}-\frac{65\!\cdots\!95}{75\!\cdots\!92}a^{3}+\frac{42\!\cdots\!55}{15\!\cdots\!84}a^{2}-\frac{11\!\cdots\!11}{75\!\cdots\!92}a-\frac{10\!\cdots\!29}{37\!\cdots\!96}$, $\frac{10\!\cdots\!47}{30\!\cdots\!68}a^{15}-\frac{93\!\cdots\!63}{30\!\cdots\!68}a^{14}+\frac{33\!\cdots\!99}{30\!\cdots\!68}a^{13}-\frac{16\!\cdots\!53}{15\!\cdots\!84}a^{12}+\frac{22\!\cdots\!81}{30\!\cdots\!68}a^{11}-\frac{22\!\cdots\!43}{37\!\cdots\!96}a^{10}-\frac{44\!\cdots\!95}{30\!\cdots\!68}a^{9}+\frac{21\!\cdots\!97}{30\!\cdots\!68}a^{8}-\frac{17\!\cdots\!65}{15\!\cdots\!84}a^{7}-\frac{50\!\cdots\!63}{15\!\cdots\!84}a^{6}+\frac{38\!\cdots\!41}{15\!\cdots\!84}a^{5}-\frac{14\!\cdots\!55}{30\!\cdots\!68}a^{4}-\frac{67\!\cdots\!69}{15\!\cdots\!84}a^{3}+\frac{86\!\cdots\!77}{30\!\cdots\!68}a^{2}-\frac{55\!\cdots\!33}{15\!\cdots\!84}a+\frac{65\!\cdots\!85}{75\!\cdots\!92}$, $\frac{46\!\cdots\!91}{67\!\cdots\!24}a^{15}-\frac{37\!\cdots\!63}{67\!\cdots\!24}a^{14}+\frac{10\!\cdots\!59}{67\!\cdots\!24}a^{13}-\frac{62\!\cdots\!57}{33\!\cdots\!12}a^{12}+\frac{84\!\cdots\!17}{67\!\cdots\!24}a^{11}+\frac{42\!\cdots\!77}{84\!\cdots\!28}a^{10}-\frac{21\!\cdots\!59}{67\!\cdots\!24}a^{9}+\frac{85\!\cdots\!73}{67\!\cdots\!24}a^{8}-\frac{36\!\cdots\!29}{33\!\cdots\!12}a^{7}-\frac{35\!\cdots\!07}{33\!\cdots\!12}a^{6}+\frac{17\!\cdots\!49}{33\!\cdots\!12}a^{5}-\frac{46\!\cdots\!51}{67\!\cdots\!24}a^{4}-\frac{59\!\cdots\!69}{33\!\cdots\!12}a^{3}+\frac{45\!\cdots\!93}{67\!\cdots\!24}a^{2}-\frac{23\!\cdots\!21}{33\!\cdots\!12}a+\frac{48\!\cdots\!61}{16\!\cdots\!56}$, $\frac{46\!\cdots\!01}{67\!\cdots\!24}a^{15}-\frac{41\!\cdots\!97}{67\!\cdots\!24}a^{14}+\frac{12\!\cdots\!45}{67\!\cdots\!24}a^{13}-\frac{64\!\cdots\!27}{33\!\cdots\!12}a^{12}+\frac{94\!\cdots\!91}{67\!\cdots\!24}a^{11}-\frac{80\!\cdots\!49}{84\!\cdots\!28}a^{10}-\frac{23\!\cdots\!65}{67\!\cdots\!24}a^{9}+\frac{97\!\cdots\!07}{67\!\cdots\!24}a^{8}-\frac{51\!\cdots\!99}{33\!\cdots\!12}a^{7}-\frac{36\!\cdots\!61}{33\!\cdots\!12}a^{6}+\frac{20\!\cdots\!35}{33\!\cdots\!12}a^{5}-\frac{59\!\cdots\!85}{67\!\cdots\!24}a^{4}-\frac{61\!\cdots\!03}{33\!\cdots\!12}a^{3}+\frac{51\!\cdots\!43}{67\!\cdots\!24}a^{2}-\frac{28\!\cdots\!19}{33\!\cdots\!12}a+\frac{55\!\cdots\!63}{16\!\cdots\!56}$, $\frac{45\!\cdots\!59}{33\!\cdots\!12}a^{15}-\frac{78\!\cdots\!67}{33\!\cdots\!12}a^{14}+\frac{34\!\cdots\!39}{33\!\cdots\!12}a^{13}-\frac{97\!\cdots\!61}{16\!\cdots\!56}a^{12}+\frac{19\!\cdots\!25}{33\!\cdots\!12}a^{11}-\frac{64\!\cdots\!65}{42\!\cdots\!64}a^{10}-\frac{29\!\cdots\!87}{33\!\cdots\!12}a^{9}+\frac{23\!\cdots\!37}{33\!\cdots\!12}a^{8}-\frac{28\!\cdots\!65}{16\!\cdots\!56}a^{7}-\frac{33\!\cdots\!03}{16\!\cdots\!56}a^{6}+\frac{45\!\cdots\!41}{16\!\cdots\!56}a^{5}-\frac{20\!\cdots\!67}{33\!\cdots\!12}a^{4}-\frac{71\!\cdots\!33}{16\!\cdots\!56}a^{3}+\frac{11\!\cdots\!97}{33\!\cdots\!12}a^{2}-\frac{73\!\cdots\!21}{16\!\cdots\!56}a+\frac{13\!\cdots\!85}{84\!\cdots\!28}$, $\frac{58\!\cdots\!95}{67\!\cdots\!24}a^{15}-\frac{25\!\cdots\!71}{67\!\cdots\!24}a^{14}-\frac{48\!\cdots\!81}{67\!\cdots\!24}a^{13}-\frac{62\!\cdots\!73}{33\!\cdots\!12}a^{12}+\frac{55\!\cdots\!61}{67\!\cdots\!24}a^{11}+\frac{58\!\cdots\!13}{84\!\cdots\!28}a^{10}-\frac{21\!\cdots\!55}{67\!\cdots\!24}a^{9}-\frac{14\!\cdots\!55}{67\!\cdots\!24}a^{8}+\frac{10\!\cdots\!39}{33\!\cdots\!12}a^{7}-\frac{36\!\cdots\!95}{33\!\cdots\!12}a^{6}+\frac{34\!\cdots\!53}{33\!\cdots\!12}a^{5}+\frac{89\!\cdots\!49}{67\!\cdots\!24}a^{4}-\frac{83\!\cdots\!05}{33\!\cdots\!12}a^{3}-\frac{37\!\cdots\!87}{67\!\cdots\!24}a^{2}+\frac{18\!\cdots\!35}{33\!\cdots\!12}a+\frac{98\!\cdots\!05}{16\!\cdots\!56}$, $\frac{96\!\cdots\!07}{67\!\cdots\!24}a^{15}-\frac{83\!\cdots\!35}{67\!\cdots\!24}a^{14}+\frac{25\!\cdots\!55}{67\!\cdots\!24}a^{13}-\frac{13\!\cdots\!37}{33\!\cdots\!12}a^{12}+\frac{19\!\cdots\!61}{67\!\cdots\!24}a^{11}+\frac{77\!\cdots\!17}{84\!\cdots\!28}a^{10}-\frac{47\!\cdots\!43}{67\!\cdots\!24}a^{9}+\frac{19\!\cdots\!25}{67\!\cdots\!24}a^{8}-\frac{10\!\cdots\!53}{33\!\cdots\!12}a^{7}-\frac{74\!\cdots\!91}{33\!\cdots\!12}a^{6}+\frac{42\!\cdots\!21}{33\!\cdots\!12}a^{5}-\frac{12\!\cdots\!39}{67\!\cdots\!24}a^{4}-\frac{13\!\cdots\!97}{33\!\cdots\!12}a^{3}+\frac{10\!\cdots\!93}{67\!\cdots\!24}a^{2}-\frac{60\!\cdots\!65}{33\!\cdots\!12}a-\frac{62\!\cdots\!63}{16\!\cdots\!56}$, $\frac{33\!\cdots\!43}{67\!\cdots\!24}a^{15}-\frac{30\!\cdots\!39}{67\!\cdots\!24}a^{14}+\frac{10\!\cdots\!55}{67\!\cdots\!24}a^{13}-\frac{48\!\cdots\!81}{33\!\cdots\!12}a^{12}+\frac{72\!\cdots\!77}{67\!\cdots\!24}a^{11}-\frac{49\!\cdots\!43}{84\!\cdots\!28}a^{10}-\frac{16\!\cdots\!99}{67\!\cdots\!24}a^{9}+\frac{76\!\cdots\!89}{67\!\cdots\!24}a^{8}-\frac{56\!\cdots\!01}{33\!\cdots\!12}a^{7}-\frac{21\!\cdots\!19}{33\!\cdots\!12}a^{6}+\frac{14\!\cdots\!09}{33\!\cdots\!12}a^{5}-\frac{54\!\cdots\!07}{67\!\cdots\!24}a^{4}-\frac{27\!\cdots\!53}{33\!\cdots\!12}a^{3}+\frac{34\!\cdots\!61}{67\!\cdots\!24}a^{2}-\frac{23\!\cdots\!21}{33\!\cdots\!12}a+\frac{25\!\cdots\!29}{16\!\cdots\!56}$
| 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: | \( 857502275.093 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{6}\cdot 857502275.093 \cdot 64}{2\cdot\sqrt{1162337480711184271576898233831313}}\cr\approx \mathstrut & 0.792352314675 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 2*x^14 - 199*x^13 + 1245*x^12 + 5773*x^11 - 51145*x^10 + 64756*x^9 + 386511*x^8 - 2205812*x^7 + 4179448*x^6 + 12113345*x^5 - 63532217*x^4 + 35405533*x^3 + 195854391*x^2 - 355933406*x + 1368484)
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 - 6*x^15 + 2*x^14 - 199*x^13 + 1245*x^12 + 5773*x^11 - 51145*x^10 + 64756*x^9 + 386511*x^8 - 2205812*x^7 + 4179448*x^6 + 12113345*x^5 - 63532217*x^4 + 35405533*x^3 + 195854391*x^2 - 355933406*x + 1368484, 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 - 6*x^15 + 2*x^14 - 199*x^13 + 1245*x^12 + 5773*x^11 - 51145*x^10 + 64756*x^9 + 386511*x^8 - 2205812*x^7 + 4179448*x^6 + 12113345*x^5 - 63532217*x^4 + 35405533*x^3 + 195854391*x^2 - 355933406*x + 1368484);
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 - 6*x^15 + 2*x^14 - 199*x^13 + 1245*x^12 + 5773*x^11 - 51145*x^10 + 64756*x^9 + 386511*x^8 - 2205812*x^7 + 4179448*x^6 + 12113345*x^5 - 63532217*x^4 + 35405533*x^3 + 195854391*x^2 - 355933406*x + 1368484);
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
$C_2^7.C_8$ (as 16T1155):
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 1024 |
| The 40 conjugacy class representatives for $C_2^7.C_8$ |
| Character table for $C_2^7.C_8$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 4.4.4913.1, 8.4.1842010303097.1 |
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]
Sibling fields
| Degree 16 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Minimal sibling: | 16.4.57681033264163530732453953.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}{,}\,{\href{/padicField/2.2.0.1}{2} }^{4}$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | $16$ | $16$ | $16$ | $16$ | $16$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(17\)
| 17.16.15.5 | $x^{16} + 17$ | $16$ | $1$ | $15$ | $C_{16}$ | $[\ ]_{16}$ |
|
\(67\)
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.2 | $x^{2} + 67$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.2.1.1 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 67.4.2.1 | $x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |