Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*x + 1)
gp: K = bnfinit(y^15 - 7*y^14 + 24*y^13 - 47*y^12 + 46*y^11 + 18*y^10 - 150*y^9 + 296*y^8 - 383*y^7 + 371*y^6 - 282*y^5 + 171*y^4 - 83*y^3 + 31*y^2 - 8*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*x + 1);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*x + 1)
\( x^{15} - 7 x^{14} + 24 x^{13} - 47 x^{12} + 46 x^{11} + 18 x^{10} - 150 x^{9} + 296 x^{8} - 383 x^{7} + \cdots + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $15$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[3, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(39671359416303616\)
\(\medspace = 2^{18}\cdot 73^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(12.78\) | 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: | $2^{3/2}73^{1/2}\approx 24.166091947189145$ | ||
| Ramified primes: |
\(2\), \(73\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $3$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$
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: |
$a^{14}-6a^{13}+18a^{12}-29a^{11}+17a^{10}+35a^{9}-115a^{8}+181a^{7}-202a^{6}+169a^{5}-113a^{4}+58a^{3}-25a^{2}+6a-1$, $a-1$, $\frac{5}{4}a^{14}-11a^{13}+\frac{173}{4}a^{12}-\frac{387}{4}a^{11}+113a^{10}+\frac{21}{4}a^{9}-\frac{1149}{4}a^{8}+\frac{2475}{4}a^{7}-\frac{3259}{4}a^{6}+\frac{1575}{2}a^{5}-\frac{1163}{2}a^{4}+\frac{1351}{4}a^{3}-152a^{2}+51a-\frac{39}{4}$, $\frac{7}{4}a^{14}-\frac{41}{4}a^{13}+30a^{12}-\frac{93}{2}a^{11}+\frac{95}{4}a^{10}+\frac{249}{4}a^{9}-\frac{755}{4}a^{8}+\frac{577}{2}a^{7}-\frac{1273}{4}a^{6}+\frac{529}{2}a^{5}-\frac{357}{2}a^{4}+\frac{381}{4}a^{3}-\frac{169}{4}a^{2}+\frac{49}{4}a-\frac{9}{4}$, $a^{14}-5a^{13}+\frac{51}{4}a^{12}-\frac{63}{4}a^{11}+\frac{7}{4}a^{10}+\frac{127}{4}a^{9}-\frac{149}{2}a^{8}+103a^{7}-\frac{439}{4}a^{6}+85a^{5}-56a^{4}+28a^{3}-12a^{2}+\frac{19}{4}a-\frac{7}{4}$, $\frac{1}{2}a^{14}-\frac{7}{2}a^{13}+\frac{41}{4}a^{12}-\frac{55}{4}a^{11}-\frac{13}{4}a^{10}+\frac{173}{4}a^{9}-\frac{155}{2}a^{8}+73a^{7}-\frac{115}{4}a^{6}-21a^{5}+\frac{105}{2}a^{4}-\frac{95}{2}a^{3}+30a^{2}-\frac{55}{4}a+\frac{17}{4}$, $\frac{21}{4}a^{14}-\frac{65}{2}a^{13}+\frac{197}{2}a^{12}-160a^{11}+\frac{367}{4}a^{10}+199a^{9}-\frac{2553}{4}a^{8}+\frac{3955}{4}a^{7}-\frac{2159}{2}a^{6}+\frac{1773}{2}a^{5}-\frac{1143}{2}a^{4}+\frac{1169}{4}a^{3}-\frac{239}{2}a^{2}+\frac{139}{4}a-6$, $a^{14}-6a^{13}+17a^{12}-24a^{11}+5a^{10}+47a^{9}-108a^{8}+141a^{7}-134a^{6}+96a^{5}-52a^{4}+23a^{3}-8a^{2}$
| 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: | \( 170.85154758 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{6}\cdot 170.85154758 \cdot 1}{2\cdot\sqrt{39671359416303616}}\cr\approx \mathstrut & 0.21111523603 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*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^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*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^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*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^15 - 7*x^14 + 24*x^13 - 47*x^12 + 46*x^11 + 18*x^10 - 150*x^9 + 296*x^8 - 383*x^7 + 371*x^6 - 282*x^5 + 171*x^4 - 83*x^3 + 31*x^2 - 8*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 60 |
| The 5 conjugacy class representatives for $A_5$ |
| Character table for $A_5$ |
Intermediate fields
| 5.1.341056.1 |
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 5 sibling: | 5.1.341056.1 |
| Degree 6 sibling: | 6.2.341056.1 |
| Degree 10 sibling: | 10.2.116319195136.1 |
| Degree 12 sibling: | deg 12 |
| Degree 20 sibling: | deg 20 |
| Degree 30 sibling: | data not computed |
| Minimal sibling: | 5.1.341056.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{3}$ | ${\href{/padicField/5.5.0.1}{5} }^{3}$ | ${\href{/padicField/7.3.0.1}{3} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{3}$ | ${\href{/padicField/13.5.0.1}{5} }^{3}$ | ${\href{/padicField/17.3.0.1}{3} }^{5}$ | ${\href{/padicField/19.3.0.1}{3} }^{5}$ | ${\href{/padicField/23.5.0.1}{5} }^{3}$ | ${\href{/padicField/29.3.0.1}{3} }^{5}$ | ${\href{/padicField/31.2.0.1}{2} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{5}$ | ${\href{/padicField/41.2.0.1}{2} }^{6}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{5}$ | ${\href{/padicField/47.3.0.1}{3} }^{5}$ | ${\href{/padicField/53.3.0.1}{3} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 2.12.18.59 | $x^{12} + 6 x^{11} + 22 x^{10} + 56 x^{9} + 126 x^{8} + 240 x^{7} + 332 x^{6} - 18 x^{5} - 459 x^{4} - 394 x^{3} - 344 x^{2} + 138 x + 423$ | $4$ | $3$ | $18$ | $A_4$ | $[2, 2]^{3}$ | |
|
\(73\)
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |