Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 2*x^11 + 5*x^10 + x^9 - 8*x^8 + 5*x^7 + 7*x^6 - 2*x^4 - 4*x - 1)
gp: K = bnfinit(y^13 - y^12 - 2*y^11 + 5*y^10 + y^9 - 8*y^8 + 5*y^7 + 7*y^6 - 2*y^4 - 4*y - 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - x^12 - 2*x^11 + 5*x^10 + x^9 - 8*x^8 + 5*x^7 + 7*x^6 - 2*x^4 - 4*x - 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - x^12 - 2*x^11 + 5*x^10 + x^9 - 8*x^8 + 5*x^7 + 7*x^6 - 2*x^4 - 4*x - 1)
\( x^{13} - x^{12} - 2x^{11} + 5x^{10} + x^{9} - 8x^{8} + 5x^{7} + 7x^{6} - 2x^{4} - 4x - 1 \)
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: | $[3, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-34372007159104\)
\(\medspace = -\,2^{6}\cdot 29\cdot 239\cdot 77487031\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.00\) | 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: | $2\cdot 29^{1/2}239^{1/2}77487031^{1/2}\approx 1465691.1159736216$ | ||
| Ramified primes: |
\(2\), \(29\), \(239\), \(77487031\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | $\Q(\sqrt{-537062611861}$) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{59}a^{11}+\frac{27}{59}a^{10}+\frac{27}{59}a^{9}+\frac{12}{59}a^{8}+\frac{1}{59}a^{7}+\frac{28}{59}a^{6}+\frac{3}{59}a^{5}-\frac{28}{59}a^{4}-\frac{15}{59}a^{3}-\frac{8}{59}a^{2}+\frac{2}{59}a-\frac{28}{59}$, $\frac{1}{3481}a^{12}+\frac{13}{3481}a^{11}+\frac{180}{3481}a^{10}-\frac{956}{3481}a^{9}+\frac{541}{3481}a^{8}+\frac{604}{3481}a^{7}+\frac{1499}{3481}a^{6}+\frac{107}{3481}a^{5}+\frac{1498}{3481}a^{4}+\frac{84}{3481}a^{3}+\frac{1176}{3481}a^{2}-\frac{941}{3481}a+\frac{746}{3481}$
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: | $7$ | 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: |
$\frac{35}{59}a^{12}-\frac{51}{59}a^{11}-\frac{46}{59}a^{10}+\frac{196}{59}a^{9}-\frac{58}{59}a^{8}-\frac{252}{59}a^{7}+\frac{301}{59}a^{6}+\frac{103}{59}a^{5}-\frac{72}{59}a^{4}-\frac{31}{59}a^{3}+\frac{14}{59}a^{2}-\frac{22}{59}a-\frac{137}{59}$, $\frac{1730}{3481}a^{12}-\frac{2054}{3481}a^{11}-\frac{3188}{3481}a^{10}+\frac{8740}{3481}a^{9}+\frac{898}{3481}a^{8}-\frac{13481}{3481}a^{7}+\frac{8893}{3481}a^{6}+\frac{10529}{3481}a^{5}-\frac{330}{3481}a^{4}-\frac{5189}{3481}a^{3}-\frac{489}{3481}a^{2}-\frac{2657}{3481}a-\frac{6358}{3481}$, $\frac{893}{3481}a^{12}-\frac{604}{3481}a^{11}-\frac{1923}{3481}a^{10}+\frac{3562}{3481}a^{9}+\frac{2381}{3481}a^{8}-\frac{5434}{3481}a^{7}+\frac{1077}{3481}a^{6}+\frac{6697}{3481}a^{5}+\frac{5317}{3481}a^{4}-\frac{2868}{3481}a^{3}-\frac{858}{3481}a^{2}+\frac{2030}{3481}a-\frac{4829}{3481}$, $\frac{23}{3481}a^{12}+\frac{358}{3481}a^{11}-\frac{1229}{3481}a^{10}+\frac{491}{3481}a^{9}+\frac{2708}{3481}a^{8}-\frac{3454}{3481}a^{7}-\frac{2162}{3481}a^{6}+\frac{6119}{3481}a^{5}-\frac{2008}{3481}a^{4}-\frac{2434}{3481}a^{3}-\frac{1272}{3481}a^{2}+\frac{2842}{3481}a-\frac{1899}{3481}$, $\frac{636}{3481}a^{12}-\frac{51}{3481}a^{11}-\frac{2222}{3481}a^{10}+\frac{2811}{3481}a^{9}+\frac{4059}{3481}a^{8}-\frac{7085}{3481}a^{7}-\frac{135}{3481}a^{6}+\frac{11766}{3481}a^{5}-\frac{1361}{3481}a^{4}-\frac{2803}{3481}a^{3}+\frac{3415}{3481}a^{2}+\frac{1023}{3481}a-\frac{2736}{3481}$, $a$, $\frac{19}{59}a^{12}-\frac{27}{59}a^{11}-\frac{25}{59}a^{10}+\frac{103}{59}a^{9}-\frac{30}{59}a^{8}-\frac{126}{59}a^{7}+\frac{159}{59}a^{6}+\frac{31}{59}a^{5}-\frac{33}{59}a^{4}+\frac{42}{59}a^{3}-\frac{8}{59}a^{2}-\frac{78}{59}a-\frac{43}{59}$
| 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: | \( 29.7815563037 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{5}\cdot 29.7815563037 \cdot 1}{2\cdot\sqrt{34372007159104}}\cr\approx \mathstrut & 0.198977639928 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 2*x^11 + 5*x^10 + x^9 - 8*x^8 + 5*x^7 + 7*x^6 - 2*x^4 - 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 - 2*x^11 + 5*x^10 + x^9 - 8*x^8 + 5*x^7 + 7*x^6 - 2*x^4 - 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 - 2*x^11 + 5*x^10 + x^9 - 8*x^8 + 5*x^7 + 7*x^6 - 2*x^4 - 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 - 2*x^11 + 5*x^10 + x^9 - 8*x^8 + 5*x^7 + 7*x^6 - 2*x^4 - 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 $S_{13}$ 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 | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.10.0.1}{10} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | R | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.5.0.1}{5} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.5 | $x^{6} + 6 x^{5} + 32 x^{4} + 90 x^{3} + 199 x^{2} + 204 x + 149$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.7.0.1 | $x^{7} + x + 1$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
|
\(29\)
| 29.2.1.1 | $x^{2} + 29$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.3.0.1 | $x^{3} + 2 x + 27$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 29.8.0.1 | $x^{8} + 3 x^{4} + 24 x^{3} + 26 x^{2} + 23 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
|
\(239\)
| $\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{239}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
|
\(77487031\)
| $\Q_{77487031}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
| Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |