Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 72*x^16 - 48*x^15 + 1539*x^14 - 2052*x^13 + 12780*x^12 - 24192*x^11 + 42615*x^10 - 74216*x^9 + 18144*x^8 + 143568*x^7 - 358539*x^6 + 677484*x^5 - 921780*x^4 + 780192*x^3 - 386640*x^2 + 103104*x - 11456)
gp: K = bnfinit(y^18 + 72*y^16 - 48*y^15 + 1539*y^14 - 2052*y^13 + 12780*y^12 - 24192*y^11 + 42615*y^10 - 74216*y^9 + 18144*y^8 + 143568*y^7 - 358539*y^6 + 677484*y^5 - 921780*y^4 + 780192*y^3 - 386640*y^2 + 103104*y - 11456, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 72*x^16 - 48*x^15 + 1539*x^14 - 2052*x^13 + 12780*x^12 - 24192*x^11 + 42615*x^10 - 74216*x^9 + 18144*x^8 + 143568*x^7 - 358539*x^6 + 677484*x^5 - 921780*x^4 + 780192*x^3 - 386640*x^2 + 103104*x - 11456);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 72*x^16 - 48*x^15 + 1539*x^14 - 2052*x^13 + 12780*x^12 - 24192*x^11 + 42615*x^10 - 74216*x^9 + 18144*x^8 + 143568*x^7 - 358539*x^6 + 677484*x^5 - 921780*x^4 + 780192*x^3 - 386640*x^2 + 103104*x - 11456)
\( x^{18} + 72 x^{16} - 48 x^{15} + 1539 x^{14} - 2052 x^{13} + 12780 x^{12} - 24192 x^{11} + 42615 x^{10} + \cdots - 11456 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[6, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(122648577847380683706253749655266134065152\)
\(\medspace = 2^{28}\cdot 3^{45}\cdot 13^{6}\cdot 179^{2}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(191.74\) | 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: | not computed | ||
| Ramified primes: |
\(2\), \(3\), \(13\), \(179\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{24}a^{9}-\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{3}{8}a-\frac{5}{12}$, $\frac{1}{24}a^{10}-\frac{1}{2}a^{4}-\frac{3}{8}a^{2}+\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{48}a^{11}-\frac{1}{48}a^{10}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{3}{16}a^{3}+\frac{17}{48}a^{2}+\frac{1}{12}a+\frac{1}{4}$, $\frac{1}{96}a^{12}-\frac{1}{96}a^{10}-\frac{1}{8}a^{8}-\frac{1}{8}a^{7}+\frac{1}{8}a^{5}-\frac{3}{32}a^{4}+\frac{11}{24}a^{3}+\frac{11}{32}a^{2}-\frac{11}{24}a-\frac{1}{8}$, $\frac{1}{192}a^{13}-\frac{1}{192}a^{12}-\frac{1}{192}a^{11}+\frac{1}{192}a^{10}-\frac{1}{48}a^{9}+\frac{1}{16}a^{7}+\frac{1}{16}a^{6}+\frac{9}{64}a^{5}-\frac{91}{192}a^{4}-\frac{11}{192}a^{3}-\frac{77}{192}a^{2}+\frac{1}{24}a-\frac{17}{48}$, $\frac{1}{1152}a^{14}-\frac{1}{576}a^{13}-\frac{1}{576}a^{12}+\frac{1}{576}a^{11}-\frac{19}{1152}a^{10}-\frac{1}{288}a^{9}-\frac{1}{96}a^{8}+\frac{1}{48}a^{7}-\frac{91}{384}a^{6}+\frac{121}{576}a^{5}-\frac{287}{576}a^{4}+\frac{19}{576}a^{3}+\frac{451}{1152}a^{2}+\frac{37}{288}a+\frac{139}{288}$, $\frac{1}{18432}a^{15}-\frac{1}{3072}a^{14}-\frac{5}{1152}a^{12}+\frac{17}{6144}a^{11}-\frac{25}{3072}a^{10}+\frac{19}{4608}a^{9}-\frac{19}{256}a^{8}+\frac{175}{2048}a^{7}+\frac{1207}{9216}a^{6}+\frac{1}{768}a^{5}+\frac{15}{32}a^{4}-\frac{3643}{18432}a^{3}+\frac{29}{3072}a^{2}-\frac{667}{1536}a-\frac{899}{2304}$, $\frac{1}{294912}a^{16}-\frac{1}{73728}a^{15}+\frac{7}{24576}a^{14}-\frac{23}{18432}a^{13}-\frac{397}{294912}a^{12}+\frac{7}{2048}a^{11}+\frac{35}{9216}a^{10}+\frac{65}{4608}a^{9}-\frac{3073}{32768}a^{8}+\frac{8015}{73728}a^{7}+\frac{133}{73728}a^{6}+\frac{1043}{6144}a^{5}-\frac{90907}{294912}a^{4}-\frac{13165}{36864}a^{3}+\frac{275}{4096}a^{2}+\frac{511}{9216}a+\frac{8269}{18432}$, $\frac{1}{4718592}a^{17}-\frac{1}{2359296}a^{16}+\frac{19}{1179648}a^{15}-\frac{25}{589824}a^{14}+\frac{1939}{4718592}a^{13}-\frac{2965}{2359296}a^{12}+\frac{385}{73728}a^{11}-\frac{287}{18432}a^{10}-\frac{7049}{4718592}a^{9}-\frac{30059}{2359296}a^{8}+\frac{34595}{1179648}a^{7}-\frac{16649}{589824}a^{6}-\frac{92155}{4718592}a^{5}-\frac{748751}{2359296}a^{4}-\frac{35759}{589824}a^{3}+\frac{84521}{294912}a^{2}-\frac{82615}{294912}a-\frac{37043}{147456}$
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
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: | $11$ | 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: |
$\frac{126268083}{262144}a^{17}+\frac{40387097}{131072}a^{16}+\frac{6857241907}{196608}a^{15}-\frac{26505183}{32768}a^{14}+\frac{194192082761}{262144}a^{13}-\frac{202313930401}{393216}a^{12}+\frac{23866405551}{4096}a^{11}-\frac{8115479349}{1024}a^{10}+\frac{12156234117023}{786432}a^{9}-\frac{3389559336189}{131072}a^{8}-\frac{511301457151}{65536}a^{7}+\frac{6307182883315}{98304}a^{6}-\frac{34513373354241}{262144}a^{5}+\frac{31733976494015}{131072}a^{4}-\frac{28422710355695}{98304}a^{3}+\frac{3127094979903}{16384}a^{2}-\frac{1051208146053}{16384}a+\frac{212056127801}{24576}$, $\frac{76471677}{524288}a^{17}+\frac{25483075}{262144}a^{16}+\frac{4154943317}{393216}a^{15}+\frac{2694027}{65536}a^{14}+\frac{117703995847}{524288}a^{13}-\frac{117710758787}{786432}a^{12}+\frac{14453147853}{8192}a^{11}-\frac{4818428619}{2048}a^{10}+\frac{7310069467985}{1572864}a^{9}-\frac{2025709960959}{262144}a^{8}-\frac{328188561257}{131072}a^{7}+\frac{3789030843665}{196608}a^{6}-\frac{20683837717647}{524288}a^{5}+\frac{19011404553133}{262144}a^{4}-\frac{16930597329241}{196608}a^{3}+\frac{1848208974021}{32768}a^{2}-\frac{616091019867}{32768}a+\frac{123222235403}{49152}$, $\frac{862279335}{524288}a^{17}+\frac{270095777}{262144}a^{16}+\frac{46816881343}{393216}a^{15}-\frac{285455271}{65536}a^{14}+\frac{1325616014309}{524288}a^{13}-\frac{1408410185617}{786432}a^{12}+\frac{162995093847}{8192}a^{11}-\frac{55957529901}{2048}a^{10}+\frac{83314726515427}{1572864}a^{9}-\frac{23298360528213}{262144}a^{8}-\frac{3386668652923}{131072}a^{7}+\frac{43241173928539}{196608}a^{6}-\frac{236922061626813}{524288}a^{5}+\frac{217877076576095}{262144}a^{4}-\frac{195690624155723}{196608}a^{3}+\frac{21613695446007}{32768}a^{2}-\frac{7296258545217}{32768}a+\frac{1478186800873}{49152}$, $\frac{88004331}{262144}a^{17}+\frac{27632513}{131072}a^{16}+\frac{4778244715}{196608}a^{15}-\frac{27922359}{32768}a^{14}+\frac{135296610161}{262144}a^{13}-\frac{143433574345}{393216}a^{12}+\frac{16634503935}{4096}a^{11}-\frac{5704871721}{1024}a^{10}+\frac{8498635848119}{786432}a^{9}-\frac{2376067834629}{131072}a^{8}-\frac{347035335703}{65536}a^{7}+\frac{4411405124587}{98304}a^{6}-\frac{24164597546985}{262144}a^{5}+\frac{22222028549783}{131072}a^{4}-\frac{19952148833351}{98304}a^{3}+\frac{2202451412823}{16384}a^{2}-\frac{742993623693}{16384}a+\frac{150413297057}{24576}$, $\frac{233620749}{524288}a^{17}+\frac{75136723}{262144}a^{16}+\frac{12688010789}{393216}a^{15}-\frac{41493093}{65536}a^{14}+\frac{359328259447}{524288}a^{13}-\frac{372385306067}{786432}a^{12}+\frac{44155835253}{8192}a^{11}-\frac{14976485175}{2048}a^{10}+\frac{22468503265057}{1572864}a^{9}-\frac{6260394543183}{262144}a^{8}-\frac{953804892281}{131072}a^{7}+\frac{11657510353217}{196608}a^{6}-\frac{63765776556735}{524288}a^{5}+\frac{58628463871453}{262144}a^{4}-\frac{52470487152937}{196608}a^{3}+\frac{5766465362805}{32768}a^{2}-\frac{1936044022347}{32768}a+\frac{390030319195}{49152}$, $\frac{44204973641975}{4718592}a^{17}+\frac{13863537460529}{2359296}a^{16}+\frac{800037385328069}{1179648}a^{15}-\frac{14322603254807}{589824}a^{14}+\frac{67\!\cdots\!21}{4718592}a^{13}-\frac{24\!\cdots\!75}{2359296}a^{12}+\frac{83\!\cdots\!87}{73728}a^{11}-\frac{28\!\cdots\!25}{18432}a^{10}+\frac{14\!\cdots\!69}{4718592}a^{9}-\frac{11\!\cdots\!25}{2359296}a^{8}-\frac{17\!\cdots\!27}{1179648}a^{7}+\frac{73\!\cdots\!33}{589824}a^{6}-\frac{12\!\cdots\!09}{4718592}a^{5}+\frac{11\!\cdots\!79}{2359296}a^{4}-\frac{33\!\cdots\!81}{589824}a^{3}+\frac{11\!\cdots\!43}{294912}a^{2}-\frac{37\!\cdots\!93}{294912}a+\frac{25\!\cdots\!75}{147456}$, $\frac{86206338238949}{4718592}a^{17}+\frac{28016225748619}{2359296}a^{16}+\frac{520272983705269}{393216}a^{15}-\frac{9987028464941}{589824}a^{14}+\frac{13\!\cdots\!95}{4718592}a^{13}-\frac{15\!\cdots\!95}{786432}a^{12}+\frac{16\!\cdots\!49}{73728}a^{11}-\frac{54\!\cdots\!71}{18432}a^{10}+\frac{91\!\cdots\!97}{1572864}a^{9}-\frac{23\!\cdots\!31}{2359296}a^{8}-\frac{35\!\cdots\!61}{1179648}a^{7}+\frac{47\!\cdots\!81}{196608}a^{6}-\frac{23\!\cdots\!99}{4718592}a^{5}+\frac{21\!\cdots\!89}{2359296}a^{4}-\frac{21\!\cdots\!33}{196608}a^{3}+\frac{21\!\cdots\!01}{294912}a^{2}-\frac{70\!\cdots\!03}{294912}a+\frac{15\!\cdots\!91}{49152}$, $\frac{24\!\cdots\!81}{4718592}a^{17}+\frac{260648524986761}{786432}a^{16}+\frac{43\!\cdots\!15}{1179648}a^{15}-\frac{271235639839645}{589824}a^{14}+\frac{12\!\cdots\!49}{1572864}a^{13}-\frac{12\!\cdots\!17}{2359296}a^{12}+\frac{45\!\cdots\!09}{73728}a^{11}-\frac{51\!\cdots\!53}{6144}a^{10}+\frac{76\!\cdots\!75}{4718592}a^{9}-\frac{64\!\cdots\!31}{2359296}a^{8}-\frac{33\!\cdots\!39}{393216}a^{7}+\frac{39\!\cdots\!71}{589824}a^{6}-\frac{65\!\cdots\!07}{4718592}a^{5}+\frac{20\!\cdots\!03}{786432}a^{4}-\frac{17\!\cdots\!83}{589824}a^{3}+\frac{58\!\cdots\!05}{294912}a^{2}-\frac{65\!\cdots\!65}{98304}a+\frac{13\!\cdots\!93}{147456}$, $\frac{5401}{4608}a^{17}+\frac{167395}{294912}a^{16}+\frac{1633795}{24576}a^{15}-\frac{895913}{73728}a^{14}+\frac{14415271}{18432}a^{13}-\frac{22728335}{32768}a^{12}+\frac{24310727}{9216}a^{11}-\frac{4807055}{1152}a^{10}-\frac{163391}{1536}a^{9}+\frac{982819973}{294912}a^{8}-\frac{1040326783}{73728}a^{7}+\frac{852008285}{24576}a^{6}-\frac{802975513}{18432}a^{5}+\frac{12717555983}{294912}a^{4}-\frac{164826481}{4096}a^{3}+\frac{978253237}{36864}a^{2}-\frac{87152153}{9216}a+\frac{8295893}{6144}$, $\frac{144998716838941}{1179648}a^{17}+\frac{39639173524039}{589824}a^{16}+\frac{26\!\cdots\!23}{294912}a^{15}-\frac{140185072451329}{147456}a^{14}+\frac{22\!\cdots\!67}{1179648}a^{13}-\frac{84\!\cdots\!17}{589824}a^{12}+\frac{27\!\cdots\!37}{18432}a^{11}-\frac{94\!\cdots\!63}{4608}a^{10}+\frac{46\!\cdots\!95}{1179648}a^{9}-\frac{37\!\cdots\!19}{589824}a^{8}-\frac{71\!\cdots\!25}{294912}a^{7}+\frac{25\!\cdots\!11}{147456}a^{6}-\frac{39\!\cdots\!43}{1179648}a^{5}+\frac{35\!\cdots\!53}{589824}a^{4}-\frac{10\!\cdots\!39}{147456}a^{3}+\frac{34\!\cdots\!37}{73728}a^{2}-\frac{11\!\cdots\!47}{73728}a+\frac{76\!\cdots\!09}{36864}$, $\frac{85\!\cdots\!19}{4718592}a^{17}+\frac{24\!\cdots\!29}{2359296}a^{16}+\frac{51\!\cdots\!83}{393216}a^{15}-\frac{71\!\cdots\!27}{589824}a^{14}+\frac{13\!\cdots\!17}{4718592}a^{13}-\frac{16\!\cdots\!37}{786432}a^{12}+\frac{16\!\cdots\!39}{73728}a^{11}-\frac{57\!\cdots\!61}{18432}a^{10}+\frac{31\!\cdots\!01}{524288}a^{9}-\frac{23\!\cdots\!97}{2359296}a^{8}-\frac{28\!\cdots\!35}{1179648}a^{7}+\frac{48\!\cdots\!15}{196608}a^{6}-\frac{24\!\cdots\!61}{4718592}a^{5}+\frac{22\!\cdots\!03}{2359296}a^{4}-\frac{22\!\cdots\!39}{196608}a^{3}+\frac{22\!\cdots\!55}{294912}a^{2}-\frac{77\!\cdots\!57}{294912}a+\frac{59\!\cdots\!43}{16384}$
| 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: | \( 5694862726710000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{6}\cdot 5694862726710000 \cdot 1}{2\cdot\sqrt{122648577847380683706253749655266134065152}}\cr\approx \mathstrut & 32.0170408461717 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 + 72*x^16 - 48*x^15 + 1539*x^14 - 2052*x^13 + 12780*x^12 - 24192*x^11 + 42615*x^10 - 74216*x^9 + 18144*x^8 + 143568*x^7 - 358539*x^6 + 677484*x^5 - 921780*x^4 + 780192*x^3 - 386640*x^2 + 103104*x - 11456)
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^18 + 72*x^16 - 48*x^15 + 1539*x^14 - 2052*x^13 + 12780*x^12 - 24192*x^11 + 42615*x^10 - 74216*x^9 + 18144*x^8 + 143568*x^7 - 358539*x^6 + 677484*x^5 - 921780*x^4 + 780192*x^3 - 386640*x^2 + 103104*x - 11456, 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^18 + 72*x^16 - 48*x^15 + 1539*x^14 - 2052*x^13 + 12780*x^12 - 24192*x^11 + 42615*x^10 - 74216*x^9 + 18144*x^8 + 143568*x^7 - 358539*x^6 + 677484*x^5 - 921780*x^4 + 780192*x^3 - 386640*x^2 + 103104*x - 11456);
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^18 + 72*x^16 - 48*x^15 + 1539*x^14 - 2052*x^13 + 12780*x^12 - 24192*x^11 + 42615*x^10 - 74216*x^9 + 18144*x^8 + 143568*x^7 - 358539*x^6 + 677484*x^5 - 921780*x^4 + 780192*x^3 - 386640*x^2 + 103104*x - 11456);
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
$C_3^6.A_4^2:C_2^2$ (as 18T891):
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 419904 |
| The 109 conjugacy class representatives for $C_3^6.A_4^2:C_2^2$ are not computed |
| Character table for $C_3^6.A_4^2:C_2^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{3}) \), 6.6.212891328.2 |
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 24 sibling: | data not computed |
| Degree 36 siblings: | 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 | R | R | $18$ | $18$ | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}$ | R | $18$ | $18$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | $18$ | ${\href{/padicField/37.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.4.9.6 | $x^{4} + 4 x^{3} + 10 x^{2} + 14$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ | |
| 2.4.9.5 | $x^{4} + 10 x^{2} + 8 x + 6$ | $4$ | $1$ | $9$ | $D_{4}$ | $[2, 3, 7/2]$ | |
|
\(3\)
| Deg $18$ | $18$ | $1$ | $45$ | |||
|
\(13\)
| 13.9.0.1 | $x^{9} + 12 x^{4} + 8 x^{3} + 12 x^{2} + 12 x + 11$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |
| 13.9.6.1 | $x^{9} + 6 x^{7} + 72 x^{6} + 12 x^{5} + 54 x^{4} - 2125 x^{3} + 288 x^{2} - 2160 x + 13928$ | $3$ | $3$ | $6$ | $C_3^2$ | $[\ ]_{3}^{3}$ | |
|
\(179\)
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{179}$ | $x + 177$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 179.3.2.1 | $x^{3} + 179$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 179.3.0.1 | $x^{3} + 4 x + 177$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 179.3.0.1 | $x^{3} + 4 x + 177$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 179.6.0.1 | $x^{6} + 7 x^{4} + 91 x^{3} + 55 x^{2} + 109 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |