Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 38*x^18 - 57*x^17 - 67*x^16 + 536*x^15 - 1218*x^14 + 1084*x^13 + 1103*x^12 - 4720*x^11 + 6327*x^10 - 2612*x^9 - 3468*x^8 + 4748*x^7 - 1187*x^6 - 1161*x^5 + 750*x^4 - 22*x^3 - 83*x^2 + 18*x - 1)
gp: K = bnfinit(y^20 - 10*y^19 + 38*y^18 - 57*y^17 - 67*y^16 + 536*y^15 - 1218*y^14 + 1084*y^13 + 1103*y^12 - 4720*y^11 + 6327*y^10 - 2612*y^9 - 3468*y^8 + 4748*y^7 - 1187*y^6 - 1161*y^5 + 750*y^4 - 22*y^3 - 83*y^2 + 18*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 38*x^18 - 57*x^17 - 67*x^16 + 536*x^15 - 1218*x^14 + 1084*x^13 + 1103*x^12 - 4720*x^11 + 6327*x^10 - 2612*x^9 - 3468*x^8 + 4748*x^7 - 1187*x^6 - 1161*x^5 + 750*x^4 - 22*x^3 - 83*x^2 + 18*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 10*x^19 + 38*x^18 - 57*x^17 - 67*x^16 + 536*x^15 - 1218*x^14 + 1084*x^13 + 1103*x^12 - 4720*x^11 + 6327*x^10 - 2612*x^9 - 3468*x^8 + 4748*x^7 - 1187*x^6 - 1161*x^5 + 750*x^4 - 22*x^3 - 83*x^2 + 18*x - 1)
\( x^{20} - 10 x^{19} + 38 x^{18} - 57 x^{17} - 67 x^{16} + 536 x^{15} - 1218 x^{14} + 1084 x^{13} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[14, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-72735891382281174868544921875\)
\(\medspace = -\,5^{10}\cdot 19^{7}\cdot 1699^{4}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(27.74\) | 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: | $5^{1/2}19^{3/4}1699^{1/2}\approx 838.7777321687686$ | ||
| Ramified primes: |
\(5\), \(19\), \(1699\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-19}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{323}a^{16}-\frac{8}{323}a^{15}+\frac{144}{323}a^{14}+\frac{101}{323}a^{13}+\frac{156}{323}a^{12}-\frac{29}{323}a^{11}+\frac{4}{323}a^{10}+\frac{161}{323}a^{9}+\frac{3}{323}a^{8}+\frac{151}{323}a^{7}-\frac{45}{323}a^{6}-\frac{93}{323}a^{5}+\frac{24}{323}a^{4}-\frac{78}{323}a^{3}+\frac{13}{323}a^{2}+\frac{141}{323}a-\frac{35}{323}$, $\frac{1}{323}a^{17}+\frac{80}{323}a^{15}-\frac{39}{323}a^{14}-\frac{5}{323}a^{13}-\frac{73}{323}a^{12}+\frac{5}{17}a^{11}-\frac{130}{323}a^{10}-\frac{1}{323}a^{9}-\frac{148}{323}a^{8}-\frac{129}{323}a^{7}-\frac{130}{323}a^{6}-\frac{74}{323}a^{5}+\frac{6}{17}a^{4}+\frac{35}{323}a^{3}-\frac{78}{323}a^{2}+\frac{124}{323}a+\frac{43}{323}$, $\frac{1}{374357}a^{18}-\frac{9}{374357}a^{17}-\frac{166}{374357}a^{16}+\frac{1532}{374357}a^{15}-\frac{10207}{374357}a^{14}+\frac{43925}{374357}a^{13}+\frac{183308}{374357}a^{12}-\frac{101087}{374357}a^{11}-\frac{38898}{374357}a^{10}-\frac{18427}{374357}a^{9}-\frac{90298}{374357}a^{8}+\frac{153163}{374357}a^{7}+\frac{30900}{374357}a^{6}-\frac{168850}{374357}a^{5}-\frac{95720}{374357}a^{4}-\frac{136568}{374357}a^{3}-\frac{122205}{374357}a^{2}-\frac{4751}{374357}a+\frac{42461}{374357}$, $\frac{1}{4117927}a^{19}-\frac{4}{4117927}a^{18}+\frac{4425}{4117927}a^{17}-\frac{457}{4117927}a^{16}+\frac{1126319}{4117927}a^{15}-\frac{354810}{4117927}a^{14}-\frac{104392}{216733}a^{13}-\frac{826850}{4117927}a^{12}+\frac{1801483}{4117927}a^{11}-\frac{10392}{22021}a^{10}-\frac{373668}{4117927}a^{9}+\frac{2006924}{4117927}a^{8}+\frac{398019}{4117927}a^{7}+\frac{2055624}{4117927}a^{6}+\frac{1445252}{4117927}a^{5}-\frac{114480}{4117927}a^{4}-\frac{2049811}{4117927}a^{3}-\frac{1741165}{4117927}a^{2}+\frac{47324}{242231}a+\frac{826575}{4117927}$
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: | $16$ | 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: |
$\frac{93605}{374357}a^{18}-\frac{842445}{374357}a^{17}+\frac{2602238}{374357}a^{16}-\frac{1722484}{374357}a^{15}-\frac{11304835}{374357}a^{14}+\frac{42443345}{374357}a^{13}-\frac{62071372}{374357}a^{12}-\frac{9467815}{374357}a^{11}+\frac{185941735}{374357}a^{10}-\frac{308752570}{374357}a^{9}+\frac{152391594}{374357}a^{8}+\frac{276418751}{374357}a^{7}-\frac{445978669}{374357}a^{6}+\frac{4179214}{22021}a^{5}+\frac{213746526}{374357}a^{4}-\frac{107622264}{374357}a^{3}-\frac{16260110}{374357}a^{2}+\frac{19338132}{374357}a-\frac{132142}{22021}$, $\frac{143346}{374357}a^{18}-\frac{1290114}{374357}a^{17}+\frac{4159644}{374357}a^{16}-\frac{4034568}{374357}a^{15}-\frac{13551619}{374357}a^{14}+\frac{63117229}{374357}a^{13}-\frac{111474218}{374357}a^{12}+\frac{44902079}{374357}a^{11}+\frac{199969022}{374357}a^{10}-\frac{471789901}{374357}a^{9}+\frac{432676488}{374357}a^{8}+\frac{50519805}{374357}a^{7}-\frac{425409256}{374357}a^{6}+\frac{228898787}{374357}a^{5}+\frac{71055650}{374357}a^{4}-\frac{82825539}{374357}a^{3}+\frac{5968633}{374357}a^{2}+\frac{8964532}{374357}a-\frac{1037630}{374357}$, $\frac{478957}{374357}a^{18}+\frac{4310613}{374357}a^{17}-\frac{14047618}{374357}a^{16}+\frac{14673716}{374357}a^{15}+\frac{42054259}{374357}a^{14}-\frac{209194545}{374357}a^{13}+\frac{386319572}{374357}a^{12}-\frac{200928233}{374357}a^{11}-\frac{594495269}{374357}a^{10}+\frac{1572930736}{374357}a^{9}-\frac{1614353741}{374357}a^{8}+\frac{9014921}{22021}a^{7}+\frac{1193066875}{374357}a^{6}-\frac{860215686}{374357}a^{5}-\frac{20369834}{374357}a^{4}+\frac{204980112}{374357}a^{3}-\frac{47242451}{374357}a^{2}-\frac{603718}{22021}a+\frac{1728754}{374357}$, $a^{19}-9a^{18}+29a^{17}-28a^{16}-95a^{15}+441a^{14}-777a^{13}+307a^{12}+1410a^{11}-3310a^{10}+3017a^{9}+405a^{8}-3063a^{7}+1685a^{6}+498a^{5}-663a^{4}+87a^{3}+65a^{2}-18a$, $\frac{236951}{374357}a^{18}-\frac{2132559}{374357}a^{17}+\frac{6761882}{374357}a^{16}-\frac{5757052}{374357}a^{15}-\frac{24856454}{374357}a^{14}+\frac{105560574}{374357}a^{13}-\frac{173545590}{374357}a^{12}+\frac{35434264}{374357}a^{11}+\frac{385910757}{374357}a^{10}-\frac{45914263}{22021}a^{9}+\frac{585068082}{374357}a^{8}+\frac{326938556}{374357}a^{7}-\frac{871387925}{374357}a^{6}+\frac{299945425}{374357}a^{5}+\frac{284802176}{374357}a^{4}-\frac{190447803}{374357}a^{3}-\frac{605381}{22021}a^{2}+\frac{28302664}{374357}a-\frac{2909687}{374357}$, $\frac{7395889}{4117927}a^{19}+\frac{70927221}{4117927}a^{18}-\frac{251409286}{4117927}a^{17}+\frac{313805561}{4117927}a^{16}+\frac{638762401}{4117927}a^{15}-\frac{3719236381}{4117927}a^{14}+\frac{7449375370}{4117927}a^{13}-\frac{4759189159}{4117927}a^{12}-\frac{10528361889}{4117927}a^{11}+\frac{2814418467}{374357}a^{10}-\frac{33817583361}{4117927}a^{9}+\frac{3987910016}{4117927}a^{8}+\frac{29447027607}{4117927}a^{7}-\frac{24547349634}{4117927}a^{6}-\frac{1241779091}{4117927}a^{5}+\frac{9228219788}{4117927}a^{4}-\frac{2807565852}{4117927}a^{3}-\frac{806171032}{4117927}a^{2}+\frac{463195309}{4117927}a-\frac{30679892}{4117927}$, $\frac{7395889}{4117927}a^{19}+\frac{69594670}{4117927}a^{18}-\frac{239416327}{4117927}a^{17}+\frac{274903929}{4117927}a^{16}+\frac{678135053}{4117927}a^{15}-\frac{3598451442}{4117927}a^{14}+\frac{6866300801}{4117927}a^{13}-\frac{3703407399}{4117927}a^{12}-\frac{11025552638}{4117927}a^{11}+\frac{2657053878}{374357}a^{10}-\frac{29475172915}{4117927}a^{9}-\frac{258836554}{4117927}a^{8}+\frac{29509021528}{4117927}a^{7}-\frac{21087383905}{4117927}a^{6}-\frac{3318438313}{4117927}a^{5}+\frac{8856854233}{4117927}a^{4}-\frac{2297494532}{4117927}a^{3}-\frac{822219207}{4117927}a^{2}+\frac{423151085}{4117927}a-\frac{41104944}{4117927}$, $a$, $\frac{3523499}{4117927}a^{19}+\frac{35471264}{4117927}a^{18}-\frac{135972620}{4117927}a^{17}+\frac{207409494}{4117927}a^{16}+\frac{229371586}{4117927}a^{15}-\frac{1906702672}{4117927}a^{14}+\frac{4382631027}{4117927}a^{13}-\frac{3991472252}{4117927}a^{12}-\frac{3771983910}{4117927}a^{11}+\frac{1526924222}{374357}a^{10}-\frac{22841980889}{4117927}a^{9}+\frac{9911218574}{4117927}a^{8}+\frac{11776478070}{4117927}a^{7}-\frac{16500116082}{4117927}a^{6}+\frac{236653999}{242231}a^{5}+\frac{3864467918}{4117927}a^{4}-\frac{2214184528}{4117927}a^{3}-\frac{75005945}{4117927}a^{2}+\frac{233606757}{4117927}a-\frac{19185652}{4117927}$, $\frac{3523499}{4117927}a^{19}+\frac{31475217}{4117927}a^{18}-\frac{100008197}{4117927}a^{17}+\frac{91292185}{4117927}a^{16}+\frac{343116470}{4117927}a^{15}-\frac{1532450266}{4117927}a^{14}+\frac{2625506273}{4117927}a^{13}-\frac{870043381}{4117927}a^{12}-\frac{5080059756}{4117927}a^{11}+\frac{1028123610}{374357}a^{10}-\frac{9706819850}{4117927}a^{9}-\frac{137246970}{242231}a^{8}+\frac{10756137276}{4117927}a^{7}-\frac{4939540462}{4117927}a^{6}-\frac{2458641554}{4117927}a^{5}+\frac{127546546}{242231}a^{4}-\frac{35155621}{4117927}a^{3}-\frac{279481832}{4117927}a^{2}+\frac{32741213}{4117927}a+\frac{188934}{4117927}$, $\frac{110246}{374357}a^{18}+\frac{992214}{374357}a^{17}-\frac{3183547}{374357}a^{16}+\frac{2978192}{374357}a^{15}+\frac{10755229}{374357}a^{14}-\frac{48689319}{374357}a^{13}+\frac{84224961}{374357}a^{12}-\frac{29253275}{374357}a^{11}-\frac{161093816}{374357}a^{10}+\frac{362318196}{374357}a^{9}-\frac{314431661}{374357}a^{8}-\frac{71701871}{374357}a^{7}+\frac{348406771}{374357}a^{6}-\frac{162980067}{374357}a^{5}-\frac{79501640}{374357}a^{4}+\frac{72222388}{374357}a^{3}-\frac{897014}{374357}a^{2}-\frac{10055495}{374357}a+\frac{423428}{374357}$, $\frac{188934}{4117927}a^{19}+\frac{2741}{12749}a^{18}-\frac{948891}{216733}a^{17}+\frac{3808999}{216733}a^{16}-\frac{101724593}{4117927}a^{15}-\frac{142564940}{4117927}a^{14}+\frac{1005444912}{4117927}a^{13}-\frac{2101989655}{4117927}a^{12}+\frac{1444415419}{4117927}a^{11}+\frac{237543528}{374357}a^{10}-\frac{8193771238}{4117927}a^{9}+\frac{9233883227}{4117927}a^{8}-\frac{1633605568}{4117927}a^{7}-\frac{335595004}{216733}a^{6}+\frac{5080080444}{4117927}a^{5}-\frac{102713968}{4117927}a^{4}-\frac{1128634354}{4117927}a^{3}+\frac{301493582}{4117927}a^{2}+\frac{30451364}{4117927}a-\frac{3466896}{4117927}$, $\frac{5648638}{4117927}a^{19}+\frac{51725126}{4117927}a^{18}-\frac{173511334}{4117927}a^{17}+\frac{196785779}{4117927}a^{16}+\frac{25313929}{216733}a^{15}-\frac{151255319}{242231}a^{14}+\frac{290814782}{242231}a^{13}-\frac{2931755034}{4117927}a^{12}-\frac{7098181201}{4117927}a^{11}+\frac{1838609932}{374357}a^{10}-\frac{21838449337}{4117927}a^{9}+\frac{3216129655}{4117927}a^{8}+\frac{16488603755}{4117927}a^{7}-\frac{828827009}{242231}a^{6}+\frac{518065149}{4117927}a^{5}+\frac{4208236468}{4117927}a^{4}-\frac{1579022712}{4117927}a^{3}-\frac{11582444}{216733}a^{2}+\frac{209656441}{4117927}a-\frac{22272174}{4117927}$, $\frac{1574708}{4117927}a^{19}+\frac{13397605}{4117927}a^{18}-\frac{37768465}{4117927}a^{17}+\frac{12652262}{4117927}a^{16}+\frac{203567310}{4117927}a^{15}-\frac{663171593}{4117927}a^{14}+\frac{808857856}{4117927}a^{13}+\frac{34112178}{242231}a^{12}-\frac{3422386622}{4117927}a^{11}+\frac{439592410}{374357}a^{10}-\frac{1094052603}{4117927}a^{9}-\frac{6683925742}{4117927}a^{8}+\frac{9022281536}{4117927}a^{7}-\frac{1704006609}{4117927}a^{6}-\frac{4537908954}{4117927}a^{5}+\frac{3127211503}{4117927}a^{4}-\frac{57635948}{4117927}a^{3}-\frac{521809374}{4117927}a^{2}+\frac{135457264}{4117927}a-\frac{3094438}{4117927}$, $\frac{37583}{242231}a^{19}-\frac{7495227}{4117927}a^{18}+\frac{35596874}{4117927}a^{17}-\frac{81184267}{4117927}a^{16}+\frac{31689919}{4117927}a^{15}+\frac{395005566}{4117927}a^{14}-\frac{1375440397}{4117927}a^{13}+\frac{2181377290}{4117927}a^{12}-\frac{895694892}{4117927}a^{11}-\frac{337709609}{374357}a^{10}+\frac{9242772458}{4117927}a^{9}-\frac{9902157774}{4117927}a^{8}+\frac{3090782859}{4117927}a^{7}+\frac{4754320452}{4117927}a^{6}-\frac{5728985582}{4117927}a^{5}+\frac{1809158291}{4117927}a^{4}+\frac{577074695}{4117927}a^{3}-\frac{528112745}{4117927}a^{2}+\frac{5704828}{216733}a-\frac{4098351}{4117927}$, $\frac{188934}{4117927}a^{19}-\frac{2995457}{4117927}a^{18}+\frac{16898271}{4117927}a^{17}-\frac{41140285}{4117927}a^{16}+\frac{14682335}{4117927}a^{15}+\frac{204772930}{4117927}a^{14}-\frac{692150218}{4117927}a^{13}+\frac{997388319}{4117927}a^{12}-\frac{78482111}{4117927}a^{11}-\frac{213156078}{374357}a^{10}+\frac{4526099332}{4117927}a^{9}-\frac{3468871508}{4117927}a^{8}-\frac{1024918082}{4117927}a^{7}+\frac{3649308819}{4117927}a^{6}-\frac{1555936295}{4117927}a^{5}-\frac{733151567}{4117927}a^{4}+\frac{565240279}{4117927}a^{3}+\frac{12654482}{4117927}a^{2}-\frac{3264033}{216733}a+\frac{5979420}{4117927}$
| 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: | \( 24265310.412863705 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{14}\cdot(2\pi)^{3}\cdot 24265310.412863705 \cdot 1}{2\cdot\sqrt{72735891382281174868544921875}}\cr\approx \mathstrut & 0.182827288086135 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 38*x^18 - 57*x^17 - 67*x^16 + 536*x^15 - 1218*x^14 + 1084*x^13 + 1103*x^12 - 4720*x^11 + 6327*x^10 - 2612*x^9 - 3468*x^8 + 4748*x^7 - 1187*x^6 - 1161*x^5 + 750*x^4 - 22*x^3 - 83*x^2 + 18*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^20 - 10*x^19 + 38*x^18 - 57*x^17 - 67*x^16 + 536*x^15 - 1218*x^14 + 1084*x^13 + 1103*x^12 - 4720*x^11 + 6327*x^10 - 2612*x^9 - 3468*x^8 + 4748*x^7 - 1187*x^6 - 1161*x^5 + 750*x^4 - 22*x^3 - 83*x^2 + 18*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^20 - 10*x^19 + 38*x^18 - 57*x^17 - 67*x^16 + 536*x^15 - 1218*x^14 + 1084*x^13 + 1103*x^12 - 4720*x^11 + 6327*x^10 - 2612*x^9 - 3468*x^8 + 4748*x^7 - 1187*x^6 - 1161*x^5 + 750*x^4 - 22*x^3 - 83*x^2 + 18*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^20 - 10*x^19 + 38*x^18 - 57*x^17 - 67*x^16 + 536*x^15 - 1218*x^14 + 1084*x^13 + 1103*x^12 - 4720*x^11 + 6327*x^10 - 2612*x^9 - 3468*x^8 + 4748*x^7 - 1187*x^6 - 1161*x^5 + 750*x^4 - 22*x^3 - 83*x^2 + 18*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
$C_2^8.D_{10}\wr C_2$ (as 20T876):
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 204800 |
| The 230 conjugacy class representatives for $C_2^8.D_{10}\wr C_2$ |
| Character table for $C_2^8.D_{10}\wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.10.3256446753125.1 |
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 20 siblings: | data not computed |
| Degree 40 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 | $20$ | $20$ | R | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{3}$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ | |
|
\(19\)
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.2 | $x^{2} + 19$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.1.1 | $x^{2} + 38$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.3.2 | $x^{4} + 38$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(1699\)
| $\Q_{1699}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{1699}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $4$ | $2$ | $2$ | $2$ | ||||
| Deg $4$ | $2$ | $2$ | $2$ |