Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*x^2 + x + 1)
gp: K = bnfinit(y^20 - 10*y^19 + 47*y^18 - 138*y^17 + 283*y^16 - 428*y^15 + 450*y^14 - 148*y^13 - 409*y^12 + 374*y^11 + 1036*y^10 - 2975*y^9 + 3655*y^8 - 2634*y^7 + 947*y^6 + 305*y^5 - 612*y^4 + 322*y^3 - 67*y^2 + y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*x^2 + x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 10*x^19 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*x^2 + x + 1)
\( x^{20} - 10 x^{19} + 47 x^{18} - 138 x^{17} + 283 x^{16} - 428 x^{15} + 450 x^{14} - 148 x^{13} + \cdots + 1 \)
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: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(33626538312268515533112249\)
\(\medspace = 3^{10}\cdot 7^{10}\cdot 17^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.89\) | 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: | $3^{1/2}7^{1/2}17^{1/2}\approx 18.894443627691185$ | ||
| Ramified primes: |
\(3\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}+\frac{1}{7}a^{11}-\frac{1}{7}a^{10}-\frac{3}{7}a^{9}-\frac{1}{7}a^{8}-\frac{2}{7}a^{7}+\frac{2}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{3}-\frac{1}{7}a^{2}+\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{13}-\frac{2}{7}a^{11}-\frac{2}{7}a^{10}+\frac{2}{7}a^{9}-\frac{1}{7}a^{8}-\frac{3}{7}a^{7}+\frac{2}{7}a^{6}+\frac{3}{7}a^{5}-\frac{1}{7}a^{4}+\frac{2}{7}a^{2}+\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{7}a^{14}+\frac{2}{7}a^{8}-\frac{2}{7}a^{7}+\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{15}+\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{16}+\frac{2}{7}a^{10}-\frac{2}{7}a^{9}+\frac{1}{7}a^{4}-\frac{2}{7}a^{3}+\frac{1}{7}a^{2}$, $\frac{1}{7}a^{17}+\frac{2}{7}a^{11}-\frac{2}{7}a^{10}+\frac{1}{7}a^{5}-\frac{2}{7}a^{4}+\frac{1}{7}a^{3}$, $\frac{1}{50699467}a^{18}-\frac{9}{50699467}a^{17}-\frac{414041}{7242781}a^{16}+\frac{1458157}{50699467}a^{15}+\frac{1723915}{50699467}a^{14}+\frac{1301}{29323}a^{13}-\frac{2248522}{50699467}a^{12}-\frac{18209069}{50699467}a^{11}+\frac{5342430}{50699467}a^{10}+\frac{6766273}{50699467}a^{9}-\frac{10902657}{50699467}a^{8}-\frac{3823174}{50699467}a^{7}-\frac{24509865}{50699467}a^{6}-\frac{19565878}{50699467}a^{5}-\frac{23275856}{50699467}a^{4}-\frac{1865545}{50699467}a^{3}+\frac{3134961}{7242781}a^{2}+\frac{2628901}{50699467}a+\frac{11121605}{50699467}$, $\frac{1}{12826965151}a^{19}+\frac{9}{986689627}a^{18}+\frac{17175408}{1832423593}a^{17}-\frac{42942520}{1166087741}a^{16}+\frac{373764003}{12826965151}a^{15}-\frac{65908385}{1832423593}a^{14}+\frac{838873669}{12826965151}a^{13}+\frac{2673961}{12826965151}a^{12}-\frac{28996712}{1166087741}a^{11}+\frac{452920397}{1166087741}a^{10}+\frac{86663040}{557694137}a^{9}+\frac{4894690390}{12826965151}a^{8}+\frac{206199527}{986689627}a^{7}+\frac{238355954}{12826965151}a^{6}-\frac{2517547608}{12826965151}a^{5}-\frac{1362939924}{12826965151}a^{4}-\frac{353265945}{1832423593}a^{3}+\frac{1449478361}{12826965151}a^{2}-\frac{84960948}{12826965151}a-\frac{635834974}{1832423593}$
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: | $11$ | 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: |
$\frac{32470775107}{12826965151}a^{19}-\frac{306528105448}{12826965151}a^{18}+\frac{1355448182122}{12826965151}a^{17}-\frac{339169302050}{1166087741}a^{16}+\frac{7138575166269}{12826965151}a^{15}-\frac{10003361593363}{12826965151}a^{14}+\frac{9201440007085}{12826965151}a^{13}+\frac{90083523941}{12826965151}a^{12}-\frac{8894921655}{8767577}a^{11}+\frac{446424596080}{1166087741}a^{10}+\frac{1565268937828}{557694137}a^{9}-\frac{76366719941186}{12826965151}a^{8}+\frac{77024114932568}{12826965151}a^{7}-\frac{2340327257448}{675103429}a^{6}+\frac{84523043369}{140955661}a^{5}+\frac{13339228358473}{12826965151}a^{4}-\frac{12406236113143}{12826965151}a^{3}+\frac{3924575846676}{12826965151}a^{2}-\frac{266382386089}{12826965151}a-\frac{56306056556}{12826965151}$, $\frac{32470775107}{12826965151}a^{19}-\frac{44345231655}{1832423593}a^{18}+\frac{1390444827355}{12826965151}a^{17}-\frac{352467065333}{1166087741}a^{16}+\frac{578117003325}{986689627}a^{15}-\frac{10670721310948}{12826965151}a^{14}+\frac{10052805705268}{12826965151}a^{13}-\frac{548189757188}{12826965151}a^{12}-\frac{1219335651336}{1166087741}a^{11}+\frac{82629715741}{166583963}a^{10}+\frac{6360484269}{2257871}a^{9}-\frac{80811977992211}{12826965151}a^{8}+\frac{84123950822165}{12826965151}a^{7}-\frac{7130933025978}{1832423593}a^{6}+\frac{9726678500909}{12826965151}a^{5}+\frac{104679632338}{96443347}a^{4}-\frac{733797636842}{675103429}a^{3}+\frac{4611324339183}{12826965151}a^{2}-\frac{232353380089}{12826965151}a-\frac{93505074632}{12826965151}$, $\frac{15285373435}{12826965151}a^{19}-\frac{136767518863}{12826965151}a^{18}+\frac{43855162077}{986689627}a^{17}-\frac{133651034406}{1166087741}a^{16}+\frac{2614239559310}{12826965151}a^{15}-\frac{3365320331627}{12826965151}a^{14}+\frac{2577192243904}{12826965151}a^{13}+\frac{204316603566}{1832423593}a^{12}-\frac{500074642359}{1166087741}a^{11}-\frac{43593718040}{1166087741}a^{10}+\frac{105249348599}{79670591}a^{9}-\frac{27562998568773}{12826965151}a^{8}+\frac{3138839801014}{1832423593}a^{7}-\frac{9066434139673}{12826965151}a^{6}-\frac{1540736737548}{12826965151}a^{5}+\frac{5674029258393}{12826965151}a^{4}-\frac{2852203561391}{12826965151}a^{3}+\frac{170873284287}{12826965151}a^{2}+\frac{7004451677}{1832423593}a+\frac{17149453812}{12826965151}$, $\frac{6434832891}{12826965151}a^{19}-\frac{69871544447}{12826965151}a^{18}+\frac{351434772725}{12826965151}a^{17}-\frac{99078024121}{1166087741}a^{16}+\frac{2333406483553}{12826965151}a^{15}-\frac{3646021639388}{12826965151}a^{14}+\frac{4010492676640}{12826965151}a^{13}-\frac{1742679559653}{12826965151}a^{12}-\frac{297939831680}{1166087741}a^{11}+\frac{399142477350}{1166087741}a^{10}+\frac{306021304507}{557694137}a^{9}-\frac{25350478641643}{12826965151}a^{8}+\frac{32943520925054}{12826965151}a^{7}-\frac{1828662293517}{986689627}a^{6}+\frac{8304119947848}{12826965151}a^{5}+\frac{3080965880989}{12826965151}a^{4}-\frac{466410196313}{986689627}a^{3}+\frac{223748715868}{986689627}a^{2}-\frac{321993911521}{12826965151}a-\frac{46370632699}{12826965151}$, $\frac{6434832891}{12826965151}a^{19}-\frac{52390280482}{12826965151}a^{18}+\frac{27729056720}{1832423593}a^{17}-\frac{2055213598}{61373039}a^{16}+\frac{617035345621}{12826965151}a^{15}-\frac{80720010845}{1832423593}a^{14}-\frac{21208801}{180661481}a^{13}+\frac{1414954613546}{12826965151}a^{12}-\frac{12599562299}{89699057}a^{11}-\frac{181121214537}{1166087741}a^{10}+\frac{297857586846}{557694137}a^{9}-\frac{5978042359708}{12826965151}a^{8}+\frac{370723979849}{12826965151}a^{7}+\frac{3127026464158}{12826965151}a^{6}-\frac{3473414038095}{12826965151}a^{5}+\frac{25849390982}{180661481}a^{4}+\frac{12674772167}{261774799}a^{3}-\frac{808125696881}{12826965151}a^{2}+\frac{98736962372}{12826965151}a+\frac{252829193}{261774799}$, $\frac{38905607998}{12826965151}a^{19}-\frac{362806902067}{12826965151}a^{18}+\frac{1584548224395}{12826965151}a^{17}-\frac{391516123695}{1166087741}a^{16}+\frac{137839938794}{217406189}a^{15}-\frac{864289337451}{986689627}a^{14}+\frac{10051299880397}{12826965151}a^{13}+\frac{866764856358}{12826965151}a^{12}-\frac{1383129961223}{1166087741}a^{11}+\frac{20909831350}{61373039}a^{10}+\frac{1868897201289}{557694137}a^{9}-\frac{86790020351919}{12826965151}a^{8}+\frac{84494674802014}{12826965151}a^{7}-\frac{46789504717688}{12826965151}a^{6}+\frac{6253264462814}{12826965151}a^{5}+\frac{15757697860676}{12826965151}a^{4}-\frac{13321091263815}{12826965151}a^{3}+\frac{3803198642302}{12826965151}a^{2}-\frac{146443382868}{12826965151}a-\frac{5253036848}{986689627}$, $\frac{13197144984}{12826965151}a^{19}-\frac{1722448027}{180661481}a^{18}+\frac{529861428009}{12826965151}a^{17}-\frac{6823070263}{61373039}a^{16}+\frac{204674327026}{986689627}a^{15}-\frac{3622296894189}{12826965151}a^{14}+\frac{3145435876150}{12826965151}a^{13}+\frac{535324019795}{12826965151}a^{12}-\frac{468593080344}{1166087741}a^{11}+\frac{100546479044}{1166087741}a^{10}+\frac{49375665373}{42899549}a^{9}-\frac{28464425591298}{12826965151}a^{8}+\frac{26546504014834}{12826965151}a^{7}-\frac{13823778621949}{12826965151}a^{6}+\frac{1084096760567}{12826965151}a^{5}+\frac{5404832489194}{12826965151}a^{4}-\frac{4075683812831}{12826965151}a^{3}+\frac{957990215326}{12826965151}a^{2}+\frac{5169308556}{12826965151}a-\frac{15285373435}{12826965151}$, $\frac{973571545}{986689627}a^{19}-\frac{16970221508}{1832423593}a^{18}+\frac{522251681316}{12826965151}a^{17}-\frac{129904244187}{1166087741}a^{16}+\frac{2716717552565}{12826965151}a^{15}-\frac{3779537668160}{12826965151}a^{14}+\frac{263577310795}{986689627}a^{13}+\frac{168800758629}{12826965151}a^{12}-\frac{457626084941}{1166087741}a^{11}+\frac{21970506603}{166583963}a^{10}+\frac{610020529892}{557694137}a^{9}-\frac{29084697921243}{12826965151}a^{8}+\frac{28806808052970}{12826965151}a^{7}-\frac{2314165486628}{1832423593}a^{6}+\frac{2361777041528}{12826965151}a^{5}+\frac{107521572035}{261774799}a^{4}-\frac{4596684130761}{12826965151}a^{3}+\frac{1330139324153}{12826965151}a^{2}-\frac{13594365544}{12826965151}a-\frac{42210836177}{12826965151}$, $\frac{9329105078}{12826965151}a^{19}-\frac{86469264393}{12826965151}a^{18}+\frac{375605226058}{12826965151}a^{17}-\frac{92412024910}{1166087741}a^{16}+\frac{273514359676}{1832423593}a^{15}-\frac{2644038244769}{12826965151}a^{14}+\frac{2365256979662}{12826965151}a^{13}+\frac{206967058458}{12826965151}a^{12}-\frac{321145904192}{1166087741}a^{11}+\frac{82152724916}{1166087741}a^{10}+\frac{23153745977}{29352323}a^{9}-\frac{2889290219743}{1832423593}a^{8}+\frac{19811469855327}{12826965151}a^{7}-\frac{11249611762261}{12826965151}a^{6}+\frac{1740114952713}{12826965151}a^{5}+\frac{14455654228}{51931033}a^{4}-\frac{162563340542}{675103429}a^{3}+\frac{1002388313428}{12826965151}a^{2}-\frac{90700735717}{12826965151}a-\frac{41423069955}{12826965151}$, $\frac{7662744996}{12826965151}a^{19}-\frac{75375422076}{12826965151}a^{18}+\frac{347116704302}{12826965151}a^{17}-\frac{90432232113}{1166087741}a^{16}+\frac{21791775308}{140955661}a^{15}-\frac{2902149490941}{12826965151}a^{14}+\frac{2890554912709}{12826965151}a^{13}-\frac{572096681488}{12826965151}a^{12}-\frac{296597928847}{1166087741}a^{11}+\frac{200288142637}{1166087741}a^{10}+\frac{28464647227}{42899549}a^{9}-\frac{3034014813352}{1832423593}a^{8}+\frac{23778537553171}{12826965151}a^{7}-\frac{15470963638257}{12826965151}a^{6}+\frac{61945300842}{180661481}a^{5}+\frac{2948844729410}{12826965151}a^{4}-\frac{3929540324158}{12826965151}a^{3}+\frac{1604200350208}{12826965151}a^{2}-\frac{256595914748}{12826965151}a+\frac{18779752751}{12826965151}$, $\frac{7662744996}{12826965151}a^{19}-\frac{70216732848}{12826965151}a^{18}+\frac{300688501250}{12826965151}a^{17}-\frac{72635431217}{1166087741}a^{16}+\frac{209900810956}{1832423593}a^{15}-\frac{1966043601607}{12826965151}a^{14}+\frac{1645062414459}{12826965151}a^{13}+\frac{456091106062}{12826965151}a^{12}-\frac{266808636909}{1166087741}a^{11}+\frac{33649229049}{1166087741}a^{10}+\frac{373353971583}{557694137}a^{9}-\frac{2245663418851}{1832423593}a^{8}+\frac{13944925496133}{12826965151}a^{7}-\frac{6736040668561}{12826965151}a^{6}+\frac{4091296232}{986689627}a^{5}+\frac{3042112730082}{12826965151}a^{4}-\frac{2044934240952}{12826965151}a^{3}+\frac{352984799931}{12826965151}a^{2}+\frac{11487800536}{12826965151}a+\frac{11193185455}{12826965151}$
| 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: | \( 58001.1435756 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 58001.1435756 \cdot 1}{2\cdot\sqrt{33626538312268515533112249}}\cr\approx \mathstrut & 0.194367840574 \end{aligned}\] (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 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*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^20 - 10*x^19 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*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^20 - 10*x^19 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*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^20 - 10*x^19 + 47*x^18 - 138*x^17 + 283*x^16 - 428*x^15 + 450*x^14 - 148*x^13 - 409*x^12 + 374*x^11 + 1036*x^10 - 2975*x^9 + 3655*x^8 - 2634*x^7 + 947*x^6 + 305*x^5 - 612*x^4 + 322*x^3 - 67*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
$C_2\times D_{10}$ (as 20T8):
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 40 |
| The 16 conjugacy class representatives for $C_2\times D_{10}$ |
| Character table for $C_2\times D_{10}$ |
Intermediate fields
| \(\Q(\sqrt{357}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{17}, \sqrt{21})\), 5.1.14161.1, 10.2.5798839393557.1, 10.2.341108199621.1, 10.2.3409076657.1 |
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
| Galois closure: | deg 40 |
| Degree 20 siblings: | 20.0.116354803848679984543641.1, 20.0.686255883923847255777801.1, deg 20 |
| Minimal sibling: | 20.0.116354803848679984543641.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/5.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{10}$ | ${\href{/padicField/13.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{10}$ | ${\href{/padicField/23.2.0.1}{2} }^{10}$ | ${\href{/padicField/29.2.0.1}{2} }^{10}$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{10}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.20.10.1 | $x^{20} + 30 x^{18} + 409 x^{16} + 4 x^{15} + 3244 x^{14} - 60 x^{13} + 16162 x^{12} - 1250 x^{11} + 53008 x^{10} - 7102 x^{9} + 121150 x^{8} - 12140 x^{7} + 219264 x^{6} + 5736 x^{5} + 257465 x^{4} + 35250 x^{3} + 250183 x^{2} + 51502 x + 77812$ | $2$ | $10$ | $10$ | 20T3 | $[\ ]_{2}^{10}$ |
|
\(7\)
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |