Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*x - 1)
gp: K = bnfinit(y^13 - y^12 - 24*y^11 + 19*y^10 + 190*y^9 - 116*y^8 - 601*y^7 + 246*y^6 + 738*y^5 - 215*y^4 - 291*y^3 + 68*y^2 + 10*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*x - 1)
\( x^{13} - x^{12} - 24 x^{11} + 19 x^{10} + 190 x^{9} - 116 x^{8} - 601 x^{7} + 246 x^{6} + 738 x^{5} + \cdots - 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $13$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[13, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(491258904256726154641\)
\(\medspace = 53^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(39.05\) | 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: | $53^{12/13}\approx 39.05163884547241$ | ||
| Ramified primes: |
\(53\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Gal(K/\Q) }$: | $13$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(53\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{53}(1,·)$, $\chi_{53}(36,·)$, $\chi_{53}(10,·)$, $\chi_{53}(44,·)$, $\chi_{53}(13,·)$, $\chi_{53}(46,·)$, $\chi_{53}(15,·)$, $\chi_{53}(16,·)$, $\chi_{53}(49,·)$, $\chi_{53}(24,·)$, $\chi_{53}(47,·)$, $\chi_{53}(28,·)$, $\chi_{53}(42,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{23}a^{11}-\frac{3}{23}a^{10}-\frac{6}{23}a^{9}-\frac{5}{23}a^{8}-\frac{10}{23}a^{7}+\frac{5}{23}a^{6}+\frac{5}{23}a^{5}-\frac{3}{23}a^{4}-\frac{1}{23}a^{3}+\frac{4}{23}a^{2}+\frac{11}{23}a+\frac{2}{23}$, $\frac{1}{435105007}a^{12}-\frac{6511450}{435105007}a^{11}-\frac{211168269}{435105007}a^{10}-\frac{202582962}{435105007}a^{9}-\frac{11741}{435105007}a^{8}+\frac{23782233}{435105007}a^{7}-\frac{11983132}{435105007}a^{6}+\frac{20632332}{435105007}a^{5}-\frac{26107826}{435105007}a^{4}+\frac{165061215}{435105007}a^{3}-\frac{53912393}{435105007}a^{2}+\frac{83591149}{435105007}a+\frac{80272926}{435105007}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: | $12$ | 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: |
$\frac{58868565}{435105007}a^{12}-\frac{57209985}{435105007}a^{11}-\frac{1394372595}{435105007}a^{10}+\frac{1053203739}{435105007}a^{9}+\frac{10743142671}{435105007}a^{8}-\frac{6041382081}{435105007}a^{7}-\frac{31985920134}{435105007}a^{6}+\frac{10933440210}{435105007}a^{5}+\frac{33861379923}{435105007}a^{4}-\frac{7816737485}{435105007}a^{3}-\frac{9306917193}{435105007}a^{2}+\frac{2392084887}{435105007}a+\frac{12930960}{435105007}$, $a$, $\frac{8620640}{435105007}a^{12}+\frac{30625070}{435105007}a^{11}-\frac{245035350}{435105007}a^{10}-\frac{765789570}{435105007}a^{9}+\frac{2340057426}{435105007}a^{8}+\frac{6162100874}{435105007}a^{7}-\frac{9208592694}{435105007}a^{6}-\frac{19203269316}{435105007}a^{5}+\frac{13650992460}{435105007}a^{4}+\frac{20720815682}{435105007}a^{3}-\frac{7574729561}{435105007}a^{2}-\frac{5618407942}{435105007}a+\frac{810719644}{435105007}$, $\frac{67072993}{435105007}a^{12}-\frac{19622855}{435105007}a^{11}-\frac{1610304692}{435105007}a^{10}+\frac{128873040}{435105007}a^{9}+\frac{12521673797}{435105007}a^{8}+\frac{1160159052}{435105007}a^{7}-\frac{37136915714}{435105007}a^{6}-\frac{433695046}{18917609}a^{5}+\frac{35880402706}{435105007}a^{4}+\frac{10172582570}{435105007}a^{3}-\frac{6595044229}{435105007}a^{2}+\frac{1068225026}{435105007}a+\frac{71383313}{435105007}$, $\frac{4310320}{435105007}a^{12}+\frac{15312535}{435105007}a^{11}-\frac{122517675}{435105007}a^{10}-\frac{382894785}{435105007}a^{9}+\frac{1170028713}{435105007}a^{8}+\frac{3081050437}{435105007}a^{7}-\frac{4604296347}{435105007}a^{6}-\frac{9601634658}{435105007}a^{5}+\frac{6825496230}{435105007}a^{4}+\frac{10360407841}{435105007}a^{3}-\frac{3569812277}{435105007}a^{2}-\frac{2809203971}{435105007}a-\frac{29745185}{435105007}$, $\frac{12600890}{435105007}a^{12}-\frac{43907430}{435105007}a^{11}-\frac{246944667}{435105007}a^{10}+\frac{950314004}{435105007}a^{9}+\frac{1256132693}{435105007}a^{8}-\frac{6568514400}{435105007}a^{7}-\frac{118932996}{435105007}a^{6}+\frac{16198493289}{435105007}a^{5}-\frac{7933636715}{435105007}a^{4}-\frac{11735438753}{435105007}a^{3}+\frac{9622726895}{435105007}a^{2}-\frac{1347519866}{435105007}a-\frac{300448027}{435105007}$, $\frac{403010}{18917609}a^{12}-\frac{6944443}{435105007}a^{11}-\frac{237391742}{435105007}a^{10}+\frac{133792468}{435105007}a^{9}+\frac{2083912137}{435105007}a^{8}-\frac{864351361}{435105007}a^{7}-\frac{7724286294}{435105007}a^{6}+\frac{1989327833}{435105007}a^{5}+\frac{11768605383}{435105007}a^{4}-\frac{1227453206}{435105007}a^{3}-\frac{6151255682}{435105007}a^{2}+\frac{332148305}{435105007}a+\frac{551832454}{435105007}$, $\frac{26027455}{435105007}a^{12}-\frac{303504}{18917609}a^{11}-\frac{630110419}{435105007}a^{10}+\frac{1836544}{18917609}a^{9}+\frac{4982037798}{435105007}a^{8}+\frac{419657564}{435105007}a^{7}-\frac{15332278548}{435105007}a^{6}-\frac{3328321358}{435105007}a^{5}+\frac{16234296775}{435105007}a^{4}+\frac{2653243924}{435105007}a^{3}-\frac{169051288}{18917609}a^{2}+\frac{1678697268}{435105007}a+\frac{166360745}{435105007}$, $\frac{17761260}{435105007}a^{12}+\frac{16549915}{435105007}a^{11}-\frac{20775770}{18917609}a^{10}-\frac{449403006}{435105007}a^{9}+\frac{191376228}{18917609}a^{8}+\frac{3693624956}{435105007}a^{7}-\frac{17132300961}{435105007}a^{6}-\frac{11149345911}{435105007}a^{5}+\frac{26784910923}{435105007}a^{4}+\frac{9603954432}{435105007}a^{3}-\frac{14227030027}{435105007}a^{2}+\frac{3181385}{435105007}a+\frac{157571832}{435105007}$, $\frac{52696650}{435105007}a^{12}-\frac{36902610}{435105007}a^{11}-\frac{1284906765}{435105007}a^{10}+\frac{635627605}{435105007}a^{9}+\frac{10393719645}{435105007}a^{8}-\frac{3415013535}{435105007}a^{7}-\frac{33851951370}{435105007}a^{6}+\frac{5741248887}{435105007}a^{5}+\frac{42533667975}{435105007}a^{4}-\frac{253155510}{18917609}a^{3}-\frac{16426724205}{435105007}a^{2}+\frac{153778200}{18917609}a-\frac{239167350}{435105007}$, $\frac{189279330}{435105007}a^{12}-\frac{194038275}{435105007}a^{11}-\frac{4521353940}{435105007}a^{10}+\frac{3708166178}{435105007}a^{9}+\frac{35469103011}{435105007}a^{8}-\frac{22872291253}{435105007}a^{7}-\frac{109989568242}{435105007}a^{6}+\frac{49863834822}{435105007}a^{5}+\frac{128440621581}{435105007}a^{4}-\frac{46580751131}{435105007}a^{3}-\frac{43014093815}{435105007}a^{2}+\frac{16800680278}{435105007}a-\frac{1051903757}{435105007}$, $\frac{116727447}{435105007}a^{12}-\frac{107458217}{435105007}a^{11}-\frac{2808403171}{435105007}a^{10}+\frac{1980429751}{435105007}a^{9}+\frac{22312007398}{435105007}a^{8}-\frac{11456471715}{435105007}a^{7}-\frac{71017547008}{435105007}a^{6}+\frac{20990665668}{435105007}a^{5}+\frac{88134183719}{435105007}a^{4}-\frac{13327795722}{435105007}a^{3}-\frac{35195140283}{435105007}a^{2}+\frac{1786210714}{435105007}a+\frac{1064317768}{435105007}$
| 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: | \( 1314145.36669 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{0}\cdot 1314145.36669 \cdot 1}{2\cdot\sqrt{491258904256726154641}}\cr\approx \mathstrut & 0.242855609085 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*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^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*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^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*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^13 - x^12 - 24*x^11 + 19*x^10 + 190*x^9 - 116*x^8 - 601*x^7 + 246*x^6 + 738*x^5 - 215*x^4 - 291*x^3 + 68*x^2 + 10*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
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 cyclic group of order 13 |
| The 13 conjugacy class representatives for $C_{13}$ |
| Character table for $C_{13}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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]
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | ${\href{/padicField/23.1.0.1}{1} }^{13}$ | ${\href{/padicField/29.13.0.1}{13} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.13.0.1}{13} }$ | ${\href{/padicField/41.13.0.1}{13} }$ | ${\href{/padicField/43.13.0.1}{13} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | R | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(53\)
| 53.13.12.1 | $x^{13} + 53$ | $13$ | $1$ | $12$ | $C_{13}$ | $[\ ]_{13}$ |