Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^12 - 91*x^10 - 74*x^9 + 2450*x^8 + 868*x^7 - 29502*x^6 + 11907*x^5 + 147462*x^4 - 165829*x^3 - 117600*x^2 + 193200*x - 56000)
gp: K = bnfinit(y^12 - 91*y^10 - 74*y^9 + 2450*y^8 + 868*y^7 - 29502*y^6 + 11907*y^5 + 147462*y^4 - 165829*y^3 - 117600*y^2 + 193200*y - 56000, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^12 - 91*x^10 - 74*x^9 + 2450*x^8 + 868*x^7 - 29502*x^6 + 11907*x^5 + 147462*x^4 - 165829*x^3 - 117600*x^2 + 193200*x - 56000);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^12 - 91*x^10 - 74*x^9 + 2450*x^8 + 868*x^7 - 29502*x^6 + 11907*x^5 + 147462*x^4 - 165829*x^3 - 117600*x^2 + 193200*x - 56000)
\( x^{12} - 91 x^{10} - 74 x^{9} + 2450 x^{8} + 868 x^{7} - 29502 x^{6} + 11907 x^{5} + 147462 x^{4} + \cdots - 56000 \)
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: |
\(16551311845247868759889\)
\(\medspace = 7^{8}\cdot 13^{6}\cdot 29^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(71.05\) | 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: | $7^{2/3}13^{1/2}29^{1/2}\approx 71.05086481765254$ | ||
| Ramified primes: |
\(7\), \(13\), \(29\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $12$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{12}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{3}a^{6}+\frac{1}{6}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}-\frac{1}{12}a^{2}+\frac{1}{12}a-\frac{1}{3}$, $\frac{1}{248820}a^{10}-\frac{1543}{49764}a^{9}-\frac{4591}{41470}a^{8}-\frac{97}{4785}a^{7}+\frac{10531}{24882}a^{6}-\frac{43601}{124410}a^{5}-\frac{15578}{62205}a^{4}-\frac{20991}{82940}a^{3}-\frac{11701}{82940}a^{2}-\frac{794}{1595}a+\frac{5185}{12441}$, $\frac{1}{98303193502800}a^{11}+\frac{2065627}{1228789918785}a^{10}+\frac{3229777417109}{98303193502800}a^{9}-\frac{647227936217}{49151596751400}a^{8}+\frac{400102106127}{3276773116760}a^{7}+\frac{506258123717}{24575798375700}a^{6}+\frac{8163633626689}{49151596751400}a^{5}+\frac{46854364408387}{98303193502800}a^{4}+\frac{9138713337491}{49151596751400}a^{3}+\frac{44586559639691}{98303193502800}a^{2}-\frac{389177936637}{819193279190}a+\frac{72028021963}{245757983757}$
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
$C_{2}\times C_{2}$, which has order $4$ (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: |
$\frac{1520283493}{98303193502800}a^{11}-\frac{166396342}{1228789918785}a^{10}-\frac{62235284121}{32767731167600}a^{9}+\frac{495180973789}{49151596751400}a^{8}+\frac{870410741149}{9830319350280}a^{7}-\frac{4563698154319}{24575798375700}a^{6}-\frac{66981617623783}{49151596751400}a^{5}+\frac{58404949584957}{32767731167600}a^{4}+\frac{124317745109141}{16383865583800}a^{3}-\frac{293170003058039}{32767731167600}a^{2}-\frac{41767374168353}{4915159675140}a+\frac{21140178335}{2824804411}$, $\frac{1520283493}{98303193502800}a^{11}-\frac{166396342}{1228789918785}a^{10}-\frac{62235284121}{32767731167600}a^{9}+\frac{495180973789}{49151596751400}a^{8}+\frac{870410741149}{9830319350280}a^{7}-\frac{4563698154319}{24575798375700}a^{6}-\frac{66981617623783}{49151596751400}a^{5}+\frac{58404949584957}{32767731167600}a^{4}+\frac{124317745109141}{16383865583800}a^{3}-\frac{293170003058039}{32767731167600}a^{2}-\frac{41767374168353}{4915159675140}a+\frac{18315373924}{2824804411}$, $\frac{58758913469}{24575798375700}a^{11}+\frac{2169232242}{409596639595}a^{10}-\frac{1266186169426}{6143949593925}a^{9}-\frac{15601660889801}{24575798375700}a^{8}+\frac{5492551646606}{1228789918785}a^{7}+\frac{74378544153973}{6143949593925}a^{6}-\frac{89894295822319}{2047983197975}a^{5}-\frac{17\!\cdots\!07}{24575798375700}a^{4}+\frac{24\!\cdots\!09}{12287899187850}a^{3}+\frac{10388004072904}{211860330825}a^{2}-\frac{865228782110531}{4915159675140}a+\frac{13955235024364}{245757983757}$, $\frac{76976693}{744721162900}a^{11}-\frac{62286907}{446832697740}a^{10}-\frac{559699106}{50776442925}a^{9}+\frac{1497298369}{372360581450}a^{8}+\frac{86404770689}{223416348870}a^{7}-\frac{4750866013}{372360581450}a^{6}-\frac{268697896252}{50776442925}a^{5}+\frac{4311844711573}{2234163488700}a^{4}+\frac{64877223996853}{2234163488700}a^{3}-\frac{26847154206893}{1117081744350}a^{2}-\frac{7705183247891}{223416348870}a+\frac{557326612075}{22341634887}$, $\frac{216934547}{558540872175}a^{11}+\frac{34272019}{89366539548}a^{10}-\frac{19093339127}{558540872175}a^{9}-\frac{12750901567}{203105771700}a^{8}+\frac{12323326011}{14894423258}a^{7}+\frac{1232906648417}{1117081744350}a^{6}-\frac{5091528010094}{558540872175}a^{5}-\frac{4244164924817}{1117081744350}a^{4}+\frac{98424960488981}{2234163488700}a^{3}-\frac{21954649349701}{1117081744350}a^{2}-\frac{1377790859881}{29788846516}a+\frac{612315492653}{22341634887}$, $\frac{6704744089}{7561784115600}a^{11}-\frac{130568741}{189044602890}a^{10}-\frac{207339675833}{2520594705200}a^{9}-\frac{55802268473}{3780892057800}a^{8}+\frac{1794739058701}{756178411560}a^{7}+\frac{19769921801}{145418925300}a^{6}-\frac{113117557919599}{3780892057800}a^{5}+\frac{26721262068601}{2520594705200}a^{4}+\frac{199354039006173}{1260297352600}a^{3}-\frac{300744478238887}{2520594705200}a^{2}-\frac{37043428582021}{189044602890}a+\frac{717473145306}{6301486763}$, $\frac{129986663629}{98303193502800}a^{11}+\frac{2724590152}{409596639595}a^{10}-\frac{3355333662513}{32767731167600}a^{9}-\frac{32350738615163}{49151596751400}a^{8}+\frac{12086966871893}{9830319350280}a^{7}+\frac{100845252323831}{8191932791900}a^{6}-\frac{110156542538959}{49151596751400}a^{5}-\frac{85\!\cdots\!57}{98303193502800}a^{4}-\frac{10\!\cdots\!21}{49151596751400}a^{3}+\frac{22\!\cdots\!19}{98303193502800}a^{2}+\frac{120864665794853}{1638386558380}a-\frac{10196411546168}{81919327919}$, $\frac{58344805763}{98303193502800}a^{11}+\frac{1663325561}{983031935028}a^{10}-\frac{1634116339111}{32767731167600}a^{9}-\frac{9176016047131}{49151596751400}a^{8}+\frac{1914088513339}{1966063870056}a^{7}+\frac{83549712155821}{24575798375700}a^{6}-\frac{424029549357113}{49151596751400}a^{5}-\frac{620639037003053}{32767731167600}a^{4}+\frac{615904802364101}{16383865583800}a^{3}+\frac{14012572679979}{1129921764400}a^{2}-\frac{8787019469489}{245757983757}a+\frac{952755599093}{81919327919}$, $\frac{637603}{177906300}a^{11}+\frac{17841}{2372084}a^{10}-\frac{54759673}{177906300}a^{9}-\frac{160547047}{177906300}a^{8}+\frac{11875286}{1779063}a^{7}+\frac{1442022077}{88953150}a^{6}-\frac{6038509853}{88953150}a^{5}-\frac{14893130029}{177906300}a^{4}+\frac{19014458037}{59302100}a^{3}-\frac{941738579}{59302100}a^{2}-\frac{2270358631}{7116252}a+\frac{261114991}{1779063}$, $\frac{49925763}{20831361200}a^{11}+\frac{16351133}{3124704180}a^{10}-\frac{13005321299}{62494083600}a^{9}-\frac{6570253541}{10415680600}a^{8}+\frac{28789639241}{6249408360}a^{7}+\frac{190588346863}{15623520900}a^{6}-\frac{1445805214199}{31247041800}a^{5}-\frac{4544084333597}{62494083600}a^{4}+\frac{2196625867723}{10415680600}a^{3}+\frac{1123141747913}{20831361200}a^{2}-\frac{204508086089}{1041568060}a+\frac{10044146579}{156235209}$, $\frac{63929499}{38778380080}a^{11}+\frac{22825717}{9694595020}a^{10}-\frac{5712440309}{38778380080}a^{9}-\frac{6344274147}{19389190040}a^{8}+\frac{69974947899}{19389190040}a^{7}+\frac{60688014873}{9694595020}a^{6}-\frac{779571515577}{19389190040}a^{5}-\frac{1186170137503}{38778380080}a^{4}+\frac{3845836433127}{19389190040}a^{3}-\frac{354938351799}{7755676016}a^{2}-\frac{509179655662}{2423648755}a+\frac{47087928231}{484729751}$
| 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: | \( 12581501.4645 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{0}\cdot 12581501.4645 \cdot 4}{2\cdot\sqrt{16551311845247868759889}}\cr\approx \mathstrut & 0.801135922040 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^12 - 91*x^10 - 74*x^9 + 2450*x^8 + 868*x^7 - 29502*x^6 + 11907*x^5 + 147462*x^4 - 165829*x^3 - 117600*x^2 + 193200*x - 56000)
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 - 91*x^10 - 74*x^9 + 2450*x^8 + 868*x^7 - 29502*x^6 + 11907*x^5 + 147462*x^4 - 165829*x^3 - 117600*x^2 + 193200*x - 56000, 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 - 91*x^10 - 74*x^9 + 2450*x^8 + 868*x^7 - 29502*x^6 + 11907*x^5 + 147462*x^4 - 165829*x^3 - 117600*x^2 + 193200*x - 56000);
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 - 91*x^10 - 74*x^9 + 2450*x^8 + 868*x^7 - 29502*x^6 + 11907*x^5 + 147462*x^4 - 165829*x^3 - 117600*x^2 + 193200*x - 56000);
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 solvable group of order 12 |
| The 4 conjugacy class representatives for $A_4$ |
| Character table for $A_4$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 4.4.6964321.1 x4, 6.6.341251729.1 x3 |
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 4 sibling: | 4.4.6964321.1 |
| Degree 6 sibling: | 6.6.341251729.1 |
| Minimal sibling: | 4.4.6964321.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{4}$ | ${\href{/padicField/3.3.0.1}{3} }^{4}$ | ${\href{/padicField/5.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{4}$ | ${\href{/padicField/23.3.0.1}{3} }^{4}$ | R | ${\href{/padicField/31.3.0.1}{3} }^{4}$ | ${\href{/padicField/37.3.0.1}{3} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}$ | ${\href{/padicField/47.3.0.1}{3} }^{4}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
| 7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
|
\(13\)
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 13.2.1.2 | $x^{2} + 26$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(29\)
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |