Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 46*x^10 + 105*x^9 + 580*x^8 - 2123*x^7 - 417*x^6 + 10028*x^5 - 14340*x^4 + 5300*x^3 + 2816*x^2 - 2156*x + 244)
gp: K = bnfinit(y^12 - y^11 - 46*y^10 + 105*y^9 + 580*y^8 - 2123*y^7 - 417*y^6 + 10028*y^5 - 14340*y^4 + 5300*y^3 + 2816*y^2 - 2156*y + 244, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - x^11 - 46*x^10 + 105*x^9 + 580*x^8 - 2123*x^7 - 417*x^6 + 10028*x^5 - 14340*x^4 + 5300*x^3 + 2816*x^2 - 2156*x + 244);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - x^11 - 46*x^10 + 105*x^9 + 580*x^8 - 2123*x^7 - 417*x^6 + 10028*x^5 - 14340*x^4 + 5300*x^3 + 2816*x^2 - 2156*x + 244)
\( x^{12} - x^{11} - 46 x^{10} + 105 x^{9} + 580 x^{8} - 2123 x^{7} - 417 x^{6} + 10028 x^{5} - 14340 x^{4} + \cdots + 244 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $12$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[12, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(26536817251562500000000\)
\(\medspace = 2^{8}\cdot 5^{14}\cdot 19^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(73.90\) | 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: | $2^{2/3}5^{13/10}19^{4/5}\approx 135.62837243568117$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{10}+\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{355010402100}a^{11}-\frac{2537619791}{118336800700}a^{10}-\frac{8275993883}{71002080420}a^{9}-\frac{1718790977}{17750520105}a^{8}+\frac{2717492636}{5916840035}a^{7}+\frac{85756288241}{177505201050}a^{6}-\frac{163103394821}{355010402100}a^{5}+\frac{8251604721}{23667360140}a^{4}+\frac{10559864407}{23667360140}a^{3}+\frac{16229827049}{35501040210}a^{2}+\frac{7660582193}{177505201050}a-\frac{80918223749}{177505201050}$
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
$C_{2}$, which has order $2$ (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: |
$\frac{59867392207}{355010402100}a^{11}-\frac{528207563}{118336800700}a^{10}+\frac{550164363431}{71002080420}a^{9}-\frac{346240030697}{35501040210}a^{8}-\frac{636790808282}{5916840035}a^{7}+\frac{21950199149069}{88752600525}a^{6}+\frac{114050520213347}{355010402100}a^{5}-\frac{32085633459477}{23667360140}a^{4}+\frac{24580569999221}{23667360140}a^{3}+\frac{4986823853287}{35501040210}a^{2}-\frac{57293587470551}{177505201050}a+\frac{7477413600893}{177505201050}$, $\frac{66572003}{118336800700}a^{11}-\frac{669157661}{29584200175}a^{10}-\frac{228053831}{5916840035}a^{9}+\frac{12328347683}{11833680070}a^{8}-\frac{11306090759}{23667360140}a^{7}-\frac{437449929101}{29584200175}a^{6}+\frac{766818917553}{29584200175}a^{5}+\frac{294140452856}{5916840035}a^{4}-\frac{3825522806117}{23667360140}a^{3}+\frac{627235368541}{5916840035}a^{2}+\frac{926262428527}{29584200175}a-\frac{2106429945997}{59168400350}$, $\frac{19216820447}{59168400350}a^{11}-\frac{6342601187}{118336800700}a^{10}-\frac{354546874437}{23667360140}a^{9}+\frac{255617928619}{11833680070}a^{8}+\frac{4880703550693}{23667360140}a^{7}-\frac{30611643503971}{59168400350}a^{6}-\frac{66823076401399}{118336800700}a^{5}+\frac{65865991862377}{23667360140}a^{4}-\frac{27750677217293}{11833680070}a^{3}-\frac{2579275031939}{11833680070}a^{2}+\frac{43091470204267}{59168400350}a-\frac{2812325365378}{29584200175}$, $\frac{13361005716}{29584200175}a^{11}+\frac{2348733193}{29584200175}a^{10}+\frac{123436009478}{5916840035}a^{9}-\frac{178506785527}{5916840035}a^{8}-\frac{1702017264942}{5916840035}a^{7}+\frac{21322609755038}{29584200175}a^{6}+\frac{23485688607561}{29584200175}a^{5}-\frac{22936032936348}{5916840035}a^{4}+\frac{19121798614004}{5916840035}a^{3}+\frac{1889959033642}{5916840035}a^{2}-\frac{29483790609176}{29584200175}a+\frac{3961882093243}{29584200175}$, $\frac{15452624657}{118336800700}a^{11}-\frac{1195847293}{59168400350}a^{10}-\frac{142549110451}{23667360140}a^{9}+\frac{203882179939}{23667360140}a^{8}+\frac{1963540392359}{23667360140}a^{7}-\frac{24477482401951}{118336800700}a^{6}-\frac{6760636238693}{29584200175}a^{5}+\frac{26330375698321}{23667360140}a^{4}-\frac{5517122118947}{5916840035}a^{3}-\frac{477073425956}{5916840035}a^{2}+\frac{16433135354501}{59168400350}a-\frac{1070178000484}{29584200175}$, $\frac{21325215558}{29584200175}a^{11}+\frac{3888830584}{29584200175}a^{10}+\frac{393815652803}{11833680070}a^{9}-\frac{573459154547}{11833680070}a^{8}-\frac{2711717961776}{5916840035}a^{7}+\frac{68377387022813}{59168400350}a^{6}+\frac{37103292469293}{29584200175}a^{5}-\frac{73513728105163}{11833680070}a^{4}+\frac{61776420571329}{11833680070}a^{3}+\frac{3029062119481}{5916840035}a^{2}-\frac{47833204776438}{29584200175}a+\frac{6249362096234}{29584200175}$, $\frac{312524884603}{177505201050}a^{11}-\frac{6939194674}{29584200175}a^{10}-\frac{5764316496703}{71002080420}a^{9}+\frac{8130907895027}{71002080420}a^{8}+\frac{6627528237396}{5916840035}a^{7}-\frac{982368238546033}{355010402100}a^{6}-\frac{555267554855663}{177505201050}a^{5}+\frac{353618385744631}{23667360140}a^{4}-\frac{291496140798743}{23667360140}a^{3}-\frac{23350646984653}{17750520105}a^{2}+\frac{677627694741583}{177505201050}a-\frac{88105908309769}{177505201050}$, $\frac{45445937653}{59168400350}a^{11}-\frac{8005488313}{118336800700}a^{10}-\frac{418325686889}{11833680070}a^{9}+\frac{1147826123777}{23667360140}a^{8}+\frac{11548298566437}{23667360140}a^{7}-\frac{140421151617983}{118336800700}a^{6}-\frac{163033213725551}{118336800700}a^{5}+\frac{38047268503422}{5916840035}a^{4}-\frac{124366841061689}{23667360140}a^{3}-\frac{6823904379791}{11833680070}a^{2}+\frac{48206748676429}{29584200175}a-\frac{12553426662919}{59168400350}$, $\frac{1053133414}{88752600525}a^{11}-\frac{849350221}{118336800700}a^{10}-\frac{20223246709}{35501040210}a^{9}+\frac{66848527867}{71002080420}a^{8}+\frac{188085173519}{23667360140}a^{7}-\frac{7300356390233}{355010402100}a^{6}-\frac{7727711666501}{355010402100}a^{5}+\frac{630989563189}{5916840035}a^{4}-\frac{2148231552663}{23667360140}a^{3}+\frac{90675111899}{35501040210}a^{2}+\frac{1966929761779}{88752600525}a-\frac{1714031821769}{177505201050}$, $\frac{39710859173}{59168400350}a^{11}+\frac{10008905183}{118336800700}a^{10}+\frac{732428461253}{23667360140}a^{9}-\frac{513993667471}{11833680070}a^{8}-\frac{10112113154277}{23667360140}a^{7}+\frac{62235002888789}{59168400350}a^{6}+\frac{141970375870391}{118336800700}a^{5}-\frac{134592849548513}{23667360140}a^{4}+\frac{55074613762127}{11833680070}a^{3}+\frac{6297587990331}{11833680070}a^{2}-\frac{85494387454303}{59168400350}a+\frac{5282011566052}{29584200175}$, $\frac{4137415808}{29584200175}a^{11}+\frac{2425805584}{29584200175}a^{10}+\frac{155020980281}{23667360140}a^{9}-\frac{279775822929}{23667360140}a^{8}-\frac{1059229877687}{11833680070}a^{7}+\frac{30573138207351}{118336800700}a^{6}+\frac{12696868494761}{59168400350}a^{5}-\frac{31883781256231}{23667360140}a^{4}+\frac{29294314154033}{23667360140}a^{3}+\frac{389383680601}{5916840035}a^{2}-\frac{22215221408601}{59168400350}a+\frac{2965764357793}{59168400350}$
| 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: | \( 57646491.00673672 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 57646491.00673672 \cdot 2}{2\cdot\sqrt{26536817251562500000000}}\cr\approx \mathstrut & 1.44946725186942 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - x^11 - 46*x^10 + 105*x^9 + 580*x^8 - 2123*x^7 - 417*x^6 + 10028*x^5 - 14340*x^4 + 5300*x^3 + 2816*x^2 - 2156*x + 244)
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^12 - x^11 - 46*x^10 + 105*x^9 + 580*x^8 - 2123*x^7 - 417*x^6 + 10028*x^5 - 14340*x^4 + 5300*x^3 + 2816*x^2 - 2156*x + 244, 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^12 - x^11 - 46*x^10 + 105*x^9 + 580*x^8 - 2123*x^7 - 417*x^6 + 10028*x^5 - 14340*x^4 + 5300*x^3 + 2816*x^2 - 2156*x + 244);
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^12 - x^11 - 46*x^10 + 105*x^9 + 580*x^8 - 2123*x^7 - 417*x^6 + 10028*x^5 - 14340*x^4 + 5300*x^3 + 2816*x^2 - 2156*x + 244);
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 60 |
| The 5 conjugacy class representatives for $A_5$ |
| Character table for $A_5$ |
Intermediate fields
| 6.6.32580250000.1 |
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 5 sibling: | data not computed |
| Degree 6 sibling: | data not computed |
| Degree 10 sibling: | data not computed |
| Degree 15 sibling: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Minimal sibling: | 5.5.8145062500.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{6}$ | ${\href{/padicField/13.3.0.1}{3} }^{4}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | ${\href{/padicField/29.3.0.1}{3} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{4}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ |
| 2.6.4.1 | $x^{6} + 3 x^{5} + 10 x^{4} + 19 x^{3} + 22 x^{2} + 11 x + 7$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
|
\(5\)
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.10.13.7 | $x^{10} + 5 x^{4} + 10$ | $10$ | $1$ | $13$ | $D_5$ | $[3/2]_{2}$ | |
|
\(19\)
| 19.2.0.1 | $x^{2} + 18 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.10.8.1 | $x^{10} + 90 x^{9} + 3250 x^{8} + 59040 x^{7} + 544360 x^{6} + 2125046 x^{5} + 1090430 x^{4} + 296960 x^{3} + 1113560 x^{2} + 9728680 x + 34800945$ | $5$ | $2$ | $8$ | $D_5$ | $[\ ]_{5}^{2}$ |