Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 + 2*x^10 + 6*x^9 - 9*x^7 - 3*x^6 + x^5 + 4*x^4 - 8*x^3 - 3*x^2 + 3*x + 1)
gp: K = bnfinit(y^13 - 3*y^12 + 2*y^10 + 6*y^9 - 9*y^7 - 3*y^6 + y^5 + 4*y^4 - 8*y^3 - 3*y^2 + 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - 3*x^12 + 2*x^10 + 6*x^9 - 9*x^7 - 3*x^6 + x^5 + 4*x^4 - 8*x^3 - 3*x^2 + 3*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - 3*x^12 + 2*x^10 + 6*x^9 - 9*x^7 - 3*x^6 + x^5 + 4*x^4 - 8*x^3 - 3*x^2 + 3*x + 1)
\( x^{13} - 3x^{12} + 2x^{10} + 6x^{9} - 9x^{7} - 3x^{6} + x^{5} + 4x^{4} - 8x^{3} - 3x^{2} + 3x + 1 \)
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: | $[5, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(1204994394291500\)
\(\medspace = 2^{2}\cdot 5^{3}\cdot 83\cdot 11171\cdot 2599231\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(14.46\) | 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: | $2\cdot 5^{3/4}83^{1/2}11171^{1/2}2599231^{1/2}\approx 10381617.17294946$ | ||
| Ramified primes: |
\(2\), \(5\), \(83\), \(11171\), \(2599231\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{12049943942915}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{25214}a^{12}+\frac{4918}{12607}a^{11}+\frac{2536}{12607}a^{10}+\frac{2452}{12607}a^{9}-\frac{4567}{12607}a^{8}-\frac{3365}{12607}a^{7}-\frac{645}{3602}a^{6}+\frac{1990}{12607}a^{5}+\frac{1879}{25214}a^{4}+\frac{5623}{25214}a^{3}+\frac{739}{3602}a^{2}-\frac{4961}{12607}a+\frac{6053}{25214}$
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 $1$
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: | $8$ | 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: |
$a$, $\frac{5967}{12607}a^{12}-\frac{19387}{12607}a^{11}+\frac{7824}{12607}a^{10}+\frac{1321}{12607}a^{9}+\frac{35304}{12607}a^{8}-\frac{4615}{12607}a^{7}-\frac{5381}{1801}a^{6}-\frac{2928}{12607}a^{5}-\frac{8237}{12607}a^{4}+\frac{17821}{12607}a^{3}-\frac{8240}{1801}a^{2}-\frac{2102}{12607}a-\frac{804}{12607}$, $\frac{5667}{25214}a^{12}-\frac{3771}{12607}a^{11}-\frac{13075}{12607}a^{10}+\frac{2570}{12607}a^{9}+\frac{26196}{12607}a^{8}+\frac{30150}{12607}a^{7}-\frac{6389}{3602}a^{6}-\frac{43756}{12607}a^{5}-\frac{42443}{25214}a^{4}+\frac{20259}{25214}a^{3}-\frac{1213}{3602}a^{2}-\frac{38198}{12607}a-\frac{13903}{25214}$, $\frac{18937}{25214}a^{12}-\frac{33564}{12607}a^{11}+\frac{16776}{12607}a^{10}+\frac{14550}{12607}a^{9}+\frac{49169}{12607}a^{8}-\frac{32441}{12607}a^{7}-\frac{21595}{3602}a^{6}+\frac{14914}{12607}a^{5}+\frac{30883}{25214}a^{4}+\frac{79671}{25214}a^{3}-\frac{31745}{3602}a^{2}+\frac{26121}{12607}a+\frac{53245}{25214}$, $\frac{24365}{25214}a^{12}-\frac{40286}{12607}a^{11}+\frac{15340}{12607}a^{10}+\frac{11014}{12607}a^{9}+\frac{70069}{12607}a^{8}-\frac{17511}{12607}a^{7}-\frac{25113}{3602}a^{6}-\frac{12779}{12607}a^{5}-\frac{6789}{25214}a^{4}+\frac{92375}{25214}a^{3}-\frac{29479}{3602}a^{2}+\frac{13758}{12607}a+\frac{29873}{25214}$, $\frac{53245}{25214}a^{12}-\frac{89336}{12607}a^{11}+\frac{33564}{12607}a^{10}+\frac{36469}{12607}a^{9}+\frac{145185}{12607}a^{8}-\frac{49169}{12607}a^{7}-\frac{59189}{3602}a^{6}-\frac{4285}{12607}a^{5}+\frac{23417}{25214}a^{4}+\frac{182097}{25214}a^{3}-\frac{72233}{3602}a^{2}+\frac{18633}{12607}a+\frac{132707}{25214}$, $\frac{28415}{25214}a^{12}-\frac{54053}{12607}a^{11}+\frac{36649}{12607}a^{10}+\frac{19905}{12607}a^{9}+\frac{68188}{12607}a^{8}-\frac{68022}{12607}a^{7}-\frac{33117}{3602}a^{6}+\frac{53883}{12607}a^{5}+\frac{64175}{25214}a^{4}+\frac{97283}{25214}a^{3}-\frac{47801}{3602}a^{2}+\frac{55087}{12607}a+\frac{112157}{25214}$, $\frac{17154}{12607}a^{12}-\frac{55772}{12607}a^{11}+\frac{16788}{12607}a^{10}+\frac{21919}{12607}a^{9}+\frac{96016}{12607}a^{8}-\frac{16728}{12607}a^{7}-\frac{18797}{1801}a^{6}-\frac{19199}{12607}a^{5}-\frac{3733}{12607}a^{4}+\frac{63820}{12607}a^{3}-\frac{22045}{1801}a^{2}-\frac{7488}{12607}a+\frac{27124}{12607}$
| 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: | \( 546.676164956 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{4}\cdot 546.676164956 \cdot 1}{2\cdot\sqrt{1204994394291500}}\cr\approx \mathstrut & 0.392714631212 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^13 - 3*x^12 + 2*x^10 + 6*x^9 - 9*x^7 - 3*x^6 + x^5 + 4*x^4 - 8*x^3 - 3*x^2 + 3*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 - 3*x^12 + 2*x^10 + 6*x^9 - 9*x^7 - 3*x^6 + x^5 + 4*x^4 - 8*x^3 - 3*x^2 + 3*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 - 3*x^12 + 2*x^10 + 6*x^9 - 9*x^7 - 3*x^6 + x^5 + 4*x^4 - 8*x^3 - 3*x^2 + 3*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 - 3*x^12 + 2*x^10 + 6*x^9 - 9*x^7 - 3*x^6 + x^5 + 4*x^4 - 8*x^3 - 3*x^2 + 3*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 non-solvable group of order 6227020800 |
| The 101 conjugacy class representatives for $S_{13}$ are not computed |
| Character table for $S_{13}$ is not computed |
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]
Sibling fields
| Degree 26 sibling: | 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 | R | ${\href{/padicField/3.7.0.1}{7} }{,}\,{\href{/padicField/3.6.0.1}{6} }$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.5.0.1}{5} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.11.0.1}{11} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.5.0.1 | $x^{5} + x^{2} + 1$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
| 2.6.0.1 | $x^{6} + x^{4} + x^{3} + x + 1$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
|
\(5\)
| 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 5.5.0.1 | $x^{5} + 4 x + 3$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
|
\(83\)
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 83.2.1.1 | $x^{2} + 166$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.0.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 83.3.0.1 | $x^{3} + 3 x + 81$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 83.5.0.1 | $x^{5} + 9 x + 81$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
|
\(11171\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(2599231\)
| $\Q_{2599231}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{2599231}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{2599231}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |