Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 23*x^13 - 66*x^12 + 140*x^11 - 211*x^10 + 217*x^9 - 270*x^8 + 455*x^7 - 685*x^6 + 904*x^5 - 975*x^4 + 826*x^3 - 521*x^2 + 251*x - 75)
gp: K = bnfinit(y^15 - 5*y^14 + 23*y^13 - 66*y^12 + 140*y^11 - 211*y^10 + 217*y^9 - 270*y^8 + 455*y^7 - 685*y^6 + 904*y^5 - 975*y^4 + 826*y^3 - 521*y^2 + 251*y - 75, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^15 - 5*x^14 + 23*x^13 - 66*x^12 + 140*x^11 - 211*x^10 + 217*x^9 - 270*x^8 + 455*x^7 - 685*x^6 + 904*x^5 - 975*x^4 + 826*x^3 - 521*x^2 + 251*x - 75);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^15 - 5*x^14 + 23*x^13 - 66*x^12 + 140*x^11 - 211*x^10 + 217*x^9 - 270*x^8 + 455*x^7 - 685*x^6 + 904*x^5 - 975*x^4 + 826*x^3 - 521*x^2 + 251*x - 75)
\( x^{15} - 5 x^{14} + 23 x^{13} - 66 x^{12} + 140 x^{11} - 211 x^{10} + 217 x^{9} - 270 x^{8} + 455 x^{7} + \cdots - 75 \)
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: | $[1, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(-23199732388265655129823\)
\(\medspace = -\,1567^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(30.98\) | 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: | $1567^{1/2}\approx 39.585350825778974$ | ||
| Ramified primes: |
\(1567\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1567}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{4}$, $\frac{1}{165}a^{13}-\frac{8}{165}a^{12}+\frac{4}{33}a^{11}-\frac{4}{33}a^{10}-\frac{2}{33}a^{9}-\frac{27}{55}a^{8}+\frac{1}{11}a^{7}+\frac{19}{55}a^{6}+\frac{4}{15}a^{5}-\frac{46}{165}a^{4}+\frac{19}{165}a^{3}-\frac{2}{33}a^{2}+\frac{13}{165}a-\frac{1}{11}$, $\frac{1}{703193681025}a^{14}-\frac{81504647}{234397893675}a^{13}-\frac{54545640691}{703193681025}a^{12}+\frac{5762743406}{140638736205}a^{11}+\frac{4934736557}{140638736205}a^{10}+\frac{34496048854}{703193681025}a^{9}+\frac{52848238666}{234397893675}a^{8}-\frac{84325384936}{234397893675}a^{7}+\frac{151155785168}{703193681025}a^{6}-\frac{10076724107}{78132631225}a^{5}+\frac{150237568387}{703193681025}a^{4}-\frac{48174930142}{703193681025}a^{3}-\frac{89186997022}{703193681025}a^{2}+\frac{150111767321}{703193681025}a-\frac{224544295}{9375915747}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
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: |
$\frac{5740101568}{63926698275}a^{14}-\frac{2908415287}{7102966475}a^{13}+\frac{115521517247}{63926698275}a^{12}-\frac{60732583357}{12785339655}a^{11}+\frac{112387786631}{12785339655}a^{10}-\frac{664408078328}{63926698275}a^{9}+\frac{122494753603}{21308899425}a^{8}-\frac{202972798948}{21308899425}a^{7}+\frac{1750381295534}{63926698275}a^{6}-\frac{795570501563}{21308899425}a^{5}+\frac{2325238824061}{63926698275}a^{4}-\frac{1912272237586}{63926698275}a^{3}+\frac{952223446154}{63926698275}a^{2}+\frac{102635746193}{63926698275}a-\frac{4056012166}{852355977}$, $\frac{11011344554}{703193681025}a^{14}-\frac{28949175028}{234397893675}a^{13}+\frac{429971634496}{703193681025}a^{12}-\frac{302673273626}{140638736205}a^{11}+\frac{770030389978}{140638736205}a^{10}-\frac{7073347254784}{703193681025}a^{9}+\frac{2968065305054}{234397893675}a^{8}-\frac{2674370002019}{234397893675}a^{7}+\frac{9157131001582}{703193681025}a^{6}-\frac{2182180214598}{78132631225}a^{5}+\frac{33297921693668}{703193681025}a^{4}-\frac{36831815114873}{703193681025}a^{3}+\frac{28670315530162}{703193681025}a^{2}-\frac{15306120958151}{703193681025}a+\frac{66221468800}{9375915747}$, $\frac{23955932866}{703193681025}a^{14}-\frac{32438403392}{234397893675}a^{13}+\frac{455067364079}{703193681025}a^{12}-\frac{226325777959}{140638736205}a^{11}+\frac{438094178117}{140638736205}a^{10}-\frac{2709703713836}{703193681025}a^{9}+\frac{684802579771}{234397893675}a^{8}-\frac{1201106784226}{234397893675}a^{7}+\frac{6407012122898}{703193681025}a^{6}-\frac{952910338757}{78132631225}a^{5}+\frac{10387852071637}{703193681025}a^{4}-\frac{9822432340177}{703193681025}a^{3}+\frac{7232974680773}{703193681025}a^{2}-\frac{3541867312849}{703193681025}a+\frac{23372690981}{9375915747}$, $\frac{31484990036}{703193681025}a^{14}-\frac{51974087317}{234397893675}a^{13}+\frac{666936357424}{703193681025}a^{12}-\frac{361054298969}{140638736205}a^{11}+\frac{653086809247}{140638736205}a^{10}-\frac{3609588204031}{703193681025}a^{9}+\frac{409738405751}{234397893675}a^{8}-\frac{579057133946}{234397893675}a^{7}+\frac{10303325460223}{703193681025}a^{6}-\frac{1748857666177}{78132631225}a^{5}+\frac{11224002361982}{703193681025}a^{4}-\frac{5057110943237}{703193681025}a^{3}+\frac{508514202208}{703193681025}a^{2}+\frac{1406193276406}{703193681025}a-\frac{4133628440}{9375915747}$, $\frac{12601406132}{703193681025}a^{14}-\frac{18831262049}{234397893675}a^{13}+\frac{249838803193}{703193681025}a^{12}-\frac{131864294933}{140638736205}a^{11}+\frac{244079933734}{140638736205}a^{10}-\frac{1534929261922}{703193681025}a^{9}+\frac{330034126157}{234397893675}a^{8}-\frac{596857041227}{234397893675}a^{7}+\frac{4044315313081}{703193681025}a^{6}-\frac{1726556540402}{234397893675}a^{5}+\frac{5924722467644}{703193681025}a^{4}-\frac{5172301927559}{703193681025}a^{3}+\frac{3275313956296}{703193681025}a^{2}-\frac{1288503577808}{703193681025}a+\frac{7535214565}{9375915747}$, $\frac{24872596894}{703193681025}a^{14}-\frac{36585609178}{234397893675}a^{13}+\frac{496223475236}{703193681025}a^{12}-\frac{259221805861}{140638736205}a^{11}+\frac{493542530168}{140638736205}a^{10}-\frac{3092560092374}{703193681025}a^{9}+\frac{697525178389}{234397893675}a^{8}-\frac{1107569756284}{234397893675}a^{7}+\frac{7327382721632}{703193681025}a^{6}-\frac{3466469734414}{234397893675}a^{5}+\frac{12506555965408}{703193681025}a^{4}-\frac{10014985275643}{703193681025}a^{3}+\frac{5002054542107}{703193681025}a^{2}-\frac{1551549715366}{703193681025}a-\frac{87728998}{9375915747}$, $\frac{13468409902}{703193681025}a^{14}-\frac{9024267713}{78132631225}a^{13}+\frac{351954340523}{703193681025}a^{12}-\frac{219739071823}{140638736205}a^{11}+\frac{453976695464}{140638736205}a^{10}-\frac{3473062218917}{703193681025}a^{9}+\frac{1046696976727}{234397893675}a^{8}-\frac{1082994646747}{234397893675}a^{7}+\frac{6801403487291}{703193681025}a^{6}-\frac{3709974672497}{234397893675}a^{5}+\frac{13990958817109}{703193681025}a^{4}-\frac{13636476281974}{703193681025}a^{3}+\frac{9836919034631}{703193681025}a^{2}-\frac{5189784772438}{703193681025}a+\frac{19178602724}{9375915747}$
| 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: | \( 404180.593485 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{7}\cdot 404180.593485 \cdot 1}{2\cdot\sqrt{23199732388265655129823}}\cr\approx \mathstrut & 1.02587169699 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 23*x^13 - 66*x^12 + 140*x^11 - 211*x^10 + 217*x^9 - 270*x^8 + 455*x^7 - 685*x^6 + 904*x^5 - 975*x^4 + 826*x^3 - 521*x^2 + 251*x - 75)
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 - 5*x^14 + 23*x^13 - 66*x^12 + 140*x^11 - 211*x^10 + 217*x^9 - 270*x^8 + 455*x^7 - 685*x^6 + 904*x^5 - 975*x^4 + 826*x^3 - 521*x^2 + 251*x - 75, 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 - 5*x^14 + 23*x^13 - 66*x^12 + 140*x^11 - 211*x^10 + 217*x^9 - 270*x^8 + 455*x^7 - 685*x^6 + 904*x^5 - 975*x^4 + 826*x^3 - 521*x^2 + 251*x - 75);
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 - 5*x^14 + 23*x^13 - 66*x^12 + 140*x^11 - 211*x^10 + 217*x^9 - 270*x^8 + 455*x^7 - 685*x^6 + 904*x^5 - 975*x^4 + 826*x^3 - 521*x^2 + 251*x - 75);
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 solvable group of order 30 |
| The 9 conjugacy class representatives for $D_{15}$ |
| Character table for $D_{15}$ |
Intermediate fields
| 3.1.1567.1, 5.1.2455489.1 |
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
| Galois closure: | deg 30 |
| 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 | $15$ | ${\href{/padicField/3.2.0.1}{2} }^{7}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.2.0.1}{2} }^{7}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }^{3}$ | ${\href{/padicField/11.2.0.1}{2} }^{7}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{7}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.2.0.1}{2} }^{7}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.2.0.1}{2} }^{7}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.2.0.1}{2} }^{7}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $15$ | ${\href{/padicField/31.3.0.1}{3} }^{5}$ | ${\href{/padicField/37.2.0.1}{2} }^{7}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15$ | $15$ | ${\href{/padicField/47.2.0.1}{2} }^{7}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.2.0.1}{2} }^{7}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $15$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(1567\)
| $\Q_{1567}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |