Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 9*x^11 + 5*x^10 + 27*x^9 - 6*x^8 - 29*x^7 - 3*x^6 + x^5 + 6*x^4 + 12*x^3 - 4*x - 1)
gp: K = bnfinit(y^13 - y^12 - 9*y^11 + 5*y^10 + 27*y^9 - 6*y^8 - 29*y^7 - 3*y^6 + y^5 + 6*y^4 + 12*y^3 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - x^12 - 9*x^11 + 5*x^10 + 27*x^9 - 6*x^8 - 29*x^7 - 3*x^6 + x^5 + 6*x^4 + 12*x^3 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - x^12 - 9*x^11 + 5*x^10 + 27*x^9 - 6*x^8 - 29*x^7 - 3*x^6 + x^5 + 6*x^4 + 12*x^3 - 4*x - 1)
\( x^{13} - x^{12} - 9x^{11} + 5x^{10} + 27x^{9} - 6x^{8} - 29x^{7} - 3x^{6} + x^{5} + 6x^{4} + 12x^{3} - 4x - 1 \)
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: | $[11, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-4161299413431551\)
\(\medspace = -\,137\cdot 14653\cdot 29411\cdot 70481\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.90\) | 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: | $137^{1/2}14653^{1/2}29411^{1/2}70481^{1/2}\approx 64508134.47489821$ | ||
| Ramified primes: |
\(137\), \(14653\), \(29411\), \(70481\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-41612\!\cdots\!31551}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Yes | |
| 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: | $11$ | 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: |
$a$, $16a^{12}-23a^{11}-134a^{10}+138a^{9}+373a^{8}-255a^{7}-357a^{6}+101a^{5}-24a^{4}+108a^{3}+145a^{2}-61a-39$, $3a^{12}-5a^{11}-24a^{10}+31a^{9}+64a^{8}-60a^{7}-59a^{6}+26a^{5}-3a^{4}+27a^{3}+22a^{2}-15a-7$, $a^{11}-a^{10}-8a^{9}+4a^{8}+19a^{7}-2a^{6}-10a^{5}-5a^{4}-9a^{3}+a^{2}+3a+1$, $15a^{12}-22a^{11}-126a^{10}+134a^{9}+354a^{8}-253a^{7}-347a^{6}+106a^{5}-15a^{4}+107a^{3}+142a^{2}-63a-39$, $3a^{12}-4a^{11}-25a^{10}+23a^{9}+68a^{8}-41a^{7}-61a^{6}+17a^{5}-10a^{4}+14a^{3}+28a^{2}-9a-6$, $7a^{12}-10a^{11}-59a^{10}+60a^{9}+167a^{8}-112a^{7}-167a^{6}+48a^{5}-4a^{4}+45a^{3}+69a^{2}-29a-19$, $5a^{12}-6a^{11}-43a^{10}+33a^{9}+121a^{8}-52a^{7}-113a^{6}+9a^{5}-15a^{4}+24a^{3}+47a^{2}-7a-9$, $4a^{12}-5a^{11}-35a^{10}+30a^{9}+102a^{8}-58a^{7}-106a^{6}+29a^{5}+a^{4}+23a^{3}+45a^{2}-19a-12$, $13a^{12}-18a^{11}-110a^{10}+107a^{9}+309a^{8}-195a^{7}-298a^{6}+75a^{5}-21a^{4}+81a^{3}+123a^{2}-46a-32$, $19a^{12}-27a^{11}-160a^{10}+163a^{9}+447a^{8}-305a^{7}-429a^{6}+125a^{5}-29a^{4}+130a^{3}+175a^{2}-74a-46$
| 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: | \( 1435.89972821 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{1}\cdot 1435.89972821 \cdot 1}{2\cdot\sqrt{4161299413431551}}\cr\approx \mathstrut & 0.143215312733 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 9*x^11 + 5*x^10 + 27*x^9 - 6*x^8 - 29*x^7 - 3*x^6 + x^5 + 6*x^4 + 12*x^3 - 4*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 - 9*x^11 + 5*x^10 + 27*x^9 - 6*x^8 - 29*x^7 - 3*x^6 + x^5 + 6*x^4 + 12*x^3 - 4*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 - 9*x^11 + 5*x^10 + 27*x^9 - 6*x^8 - 29*x^7 - 3*x^6 + x^5 + 6*x^4 + 12*x^3 - 4*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 - 9*x^11 + 5*x^10 + 27*x^9 - 6*x^8 - 29*x^7 - 3*x^6 + x^5 + 6*x^4 + 12*x^3 - 4*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 $S_{13}$ |
| Character table for $S_{13}$ |
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]
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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.13.0.1}{13} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.13.0.1}{13} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | ${\href{/padicField/23.11.0.1}{11} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.4.0.1}{4} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.13.0.1}{13} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.13.0.1}{13} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(137\)
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 137.4.0.1 | $x^{4} + x^{2} + 95 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 137.4.0.1 | $x^{4} + x^{2} + 95 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
|
\(14653\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(29411\)
| $\Q_{29411}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
|
\(70481\)
| $\Q_{70481}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |