Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 50*x^11 + 25*x^10 + 722*x^9 - 226*x^8 - 4207*x^7 + 1158*x^6 + 10465*x^5 - 2535*x^4 - 9399*x^3 + 1079*x^2 + 1316*x - 49)
gp: K = bnfinit(y^13 - y^12 - 50*y^11 + 25*y^10 + 722*y^9 - 226*y^8 - 4207*y^7 + 1158*y^6 + 10465*y^5 - 2535*y^4 - 9399*y^3 + 1079*y^2 + 1316*y - 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^13 - x^12 - 50*x^11 + 25*x^10 + 722*x^9 - 226*x^8 - 4207*x^7 + 1158*x^6 + 10465*x^5 - 2535*x^4 - 9399*x^3 + 1079*x^2 + 1316*x - 49);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^13 - x^12 - 50*x^11 + 25*x^10 + 722*x^9 - 226*x^8 - 4207*x^7 + 1158*x^6 + 10465*x^5 - 2535*x^4 - 9399*x^3 + 1079*x^2 + 1316*x - 49)
\( x^{13} - x^{12} - 50 x^{11} + 25 x^{10} + 722 x^{9} - 226 x^{8} - 4207 x^{7} + 1158 x^{6} + 10465 x^{5} + \cdots - 49 \)
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: | $[13, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(361712403654141025034641\)
\(\medspace = 23^{6}\cdot 367^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(64.89\) | 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: | $23^{1/2}367^{1/2}\approx 91.87491496594704$ | ||
| Ramified primes: |
\(23\), \(367\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a$, $\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{9}a^{8}-\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{2}{9}a^{4}+\frac{4}{9}a^{3}-\frac{2}{9}a^{2}-\frac{2}{9}a+\frac{1}{9}$, $\frac{1}{4347}a^{11}-\frac{236}{4347}a^{10}+\frac{152}{4347}a^{9}+\frac{418}{4347}a^{8}+\frac{212}{4347}a^{7}-\frac{5}{189}a^{6}-\frac{1346}{4347}a^{5}-\frac{169}{1449}a^{4}+\frac{83}{483}a^{3}+\frac{47}{207}a^{2}+\frac{75}{161}a-\frac{37}{621}$, $\frac{1}{763711828047}a^{12}-\frac{1230811}{11068287363}a^{11}-\frac{7031901503}{763711828047}a^{10}-\frac{16168789291}{109101689721}a^{9}-\frac{75055839113}{763711828047}a^{8}-\frac{2808875978}{84856869783}a^{7}+\frac{43367213471}{763711828047}a^{6}-\frac{125553895993}{763711828047}a^{5}-\frac{15307051937}{254570609349}a^{4}+\frac{86906477276}{254570609349}a^{3}+\frac{31678933072}{254570609349}a^{2}+\frac{5257070608}{33204862089}a-\frac{14192619422}{109101689721}$
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 $1$ (assuming GRH)
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: | $12$ | 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: |
$\frac{5499470}{1229809707}a^{12}+\frac{14538988}{9428541087}a^{11}-\frac{690139790}{3142847029}a^{10}-\frac{550029331}{3142847029}a^{9}+\frac{9053955502}{3142847029}a^{8}+\frac{69451528081}{28285623261}a^{7}-\frac{2407734562}{175687101}a^{6}-\frac{250958876704}{28285623261}a^{5}+\frac{98975968853}{4040803323}a^{4}+\frac{235948857290}{28285623261}a^{3}-\frac{274488541234}{28285623261}a^{2}-\frac{14179386248}{28285623261}a-\frac{712207484}{1346934441}$, $\frac{1504961978}{763711828047}a^{12}-\frac{523422649}{254570609349}a^{11}-\frac{71436103372}{763711828047}a^{10}+\frac{1576846367}{33204862089}a^{9}+\frac{911359853261}{763711828047}a^{8}-\frac{29361945518}{84856869783}a^{7}-\frac{4232586116630}{763711828047}a^{6}+\frac{903319383982}{763711828047}a^{5}+\frac{2451784299455}{254570609349}a^{4}+\frac{7550374084}{36367229907}a^{3}-\frac{1681809838864}{254570609349}a^{2}-\frac{147807171212}{109101689721}a+\frac{50976497693}{109101689721}$, $\frac{6895622612}{763711828047}a^{12}-\frac{1380367432}{254570609349}a^{11}-\frac{48774814999}{109101689721}a^{10}+\frac{39628418884}{763711828047}a^{9}+\frac{677984076614}{109101689721}a^{8}+\frac{1758274088}{28285623261}a^{7}-\frac{25694593336124}{763711828047}a^{6}+\frac{1806497771506}{763711828047}a^{5}+\frac{19011916084418}{254570609349}a^{4}-\frac{3959890368572}{254570609349}a^{3}-\frac{13471312347460}{254570609349}a^{2}+\frac{13196245272421}{763711828047}a+\frac{14210536454}{109101689721}$, $\frac{163999282}{254570609349}a^{12}-\frac{16343714}{28285623261}a^{11}-\frac{7386180356}{254570609349}a^{10}+\frac{3535325765}{254570609349}a^{9}+\frac{3346322036}{11068287363}a^{8}-\frac{17837816687}{84856869783}a^{7}-\frac{117964107400}{254570609349}a^{6}+\frac{64608636131}{36367229907}a^{5}-\frac{366558343454}{84856869783}a^{4}-\frac{315201559219}{84856869783}a^{3}+\frac{1079689363603}{84856869783}a^{2}-\frac{456738211423}{254570609349}a-\frac{92761442936}{36367229907}$, $\frac{1360420196}{763711828047}a^{12}-\frac{2164374853}{254570609349}a^{11}-\frac{66255696424}{763711828047}a^{10}+\frac{40449048766}{109101689721}a^{9}+\frac{1018321666433}{763711828047}a^{8}-\frac{43968495052}{9428541087}a^{7}-\frac{6563539952534}{763711828047}a^{6}+\frac{16862903887072}{763711828047}a^{5}+\frac{5479197251636}{254570609349}a^{4}-\frac{9321142124498}{254570609349}a^{3}-\frac{4049709745327}{254570609349}a^{2}+\frac{4861765865008}{763711828047}a+\frac{242513118731}{109101689721}$, $\frac{228778247}{84856869783}a^{12}+\frac{30062200}{12122409969}a^{11}-\frac{166732214}{1229809707}a^{10}-\frac{5026287713}{28285623261}a^{9}+\frac{52134330013}{28285623261}a^{8}+\frac{29264387552}{12122409969}a^{7}-\frac{87685651484}{9428541087}a^{6}-\frac{867070862653}{84856869783}a^{5}+\frac{549661466776}{28285623261}a^{4}+\frac{429660467791}{28285623261}a^{3}-\frac{451223930072}{28285623261}a^{2}-\frac{466841448527}{84856869783}a+\frac{79694831074}{12122409969}$, $\frac{189810361}{33204862089}a^{12}+\frac{1101851612}{254570609349}a^{11}-\frac{216242084845}{763711828047}a^{10}-\frac{267574496234}{763711828047}a^{9}+\frac{2867120394785}{763711828047}a^{8}+\frac{443008443664}{84856869783}a^{7}-\frac{605397002278}{33204862089}a^{6}-\frac{19042376306990}{763711828047}a^{5}+\frac{8702315660009}{254570609349}a^{4}+\frac{1595338662256}{36367229907}a^{3}-\frac{4455337996402}{254570609349}a^{2}-\frac{2501171254031}{109101689721}a+\frac{89902610480}{109101689721}$, $\frac{8348499212}{763711828047}a^{12}+\frac{3321573176}{254570609349}a^{11}-\frac{412024986577}{763711828047}a^{10}-\frac{658333720787}{763711828047}a^{9}+\frac{5290938979397}{763711828047}a^{8}+\frac{316642900271}{28285623261}a^{7}-\frac{23715968006648}{763711828047}a^{6}-\frac{1421453475595}{33204862089}a^{5}+\frac{13120340650139}{254570609349}a^{4}+\frac{1791541905298}{36367229907}a^{3}-\frac{5141313893977}{254570609349}a^{2}-\frac{263532438914}{109101689721}a+\frac{294882472289}{109101689721}$, $\frac{121913912}{33204862089}a^{12}-\frac{145560851}{254570609349}a^{11}-\frac{134810055992}{763711828047}a^{10}-\frac{48961296676}{763711828047}a^{9}+\frac{1724861418955}{763711828047}a^{8}+\frac{98646050098}{84856869783}a^{7}-\frac{355844452832}{33204862089}a^{6}-\frac{449545036423}{109101689721}a^{5}+\frac{5371191666733}{254570609349}a^{4}+\frac{726229515356}{254570609349}a^{3}-\frac{3830388396521}{254570609349}a^{2}+\frac{596655848468}{763711828047}a+\frac{182138669050}{109101689721}$, $\frac{13467318391}{763711828047}a^{12}-\frac{5802826514}{254570609349}a^{11}-\frac{644115803978}{763711828047}a^{10}+\frac{517441346963}{763711828047}a^{9}+\frac{8415643756831}{763711828047}a^{8}-\frac{211904075777}{28285623261}a^{7}-\frac{40360229666752}{763711828047}a^{6}+\frac{4567882994204}{109101689721}a^{5}+\frac{22003153564459}{254570609349}a^{4}-\frac{21488823891694}{254570609349}a^{3}-\frac{2057697541229}{254570609349}a^{2}+\frac{9387477129584}{763711828047}a-\frac{45098823317}{109101689721}$, $\frac{2836007764}{763711828047}a^{12}+\frac{1561544155}{254570609349}a^{11}-\frac{133139166728}{763711828047}a^{10}-\frac{42512592889}{109101689721}a^{9}+\frac{1467129001486}{763711828047}a^{8}+\frac{49476530923}{9428541087}a^{7}-\frac{4371863148343}{763711828047}a^{6}-\frac{16670252533804}{763711828047}a^{5}+\frac{298057834567}{254570609349}a^{4}+\frac{7486928073281}{254570609349}a^{3}+\frac{2138368279753}{254570609349}a^{2}-\frac{4699866177850}{763711828047}a+\frac{30408818113}{109101689721}$, $\frac{34308734458}{763711828047}a^{12}+\frac{6928032967}{254570609349}a^{11}-\frac{1682243914040}{763711828047}a^{10}-\frac{1838509304935}{763711828047}a^{9}+\frac{21820650899842}{763711828047}a^{8}+\frac{334118737438}{9428541087}a^{7}-\frac{100840301972485}{763711828047}a^{6}-\frac{17113625003833}{109101689721}a^{5}+\frac{55572170538391}{254570609349}a^{4}+\frac{57351305907635}{254570609349}a^{3}-\frac{16193039738042}{254570609349}a^{2}-\frac{30832464767707}{763711828047}a+\frac{167002148185}{109101689721}$
| 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: | \( 100827679.942 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{13}\cdot(2\pi)^{0}\cdot 100827679.942 \cdot 1}{2\cdot\sqrt{361712403654141025034641}}\cr\approx \mathstrut & 0.686685728207 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^13 - x^12 - 50*x^11 + 25*x^10 + 722*x^9 - 226*x^8 - 4207*x^7 + 1158*x^6 + 10465*x^5 - 2535*x^4 - 9399*x^3 + 1079*x^2 + 1316*x - 49)
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 - 50*x^11 + 25*x^10 + 722*x^9 - 226*x^8 - 4207*x^7 + 1158*x^6 + 10465*x^5 - 2535*x^4 - 9399*x^3 + 1079*x^2 + 1316*x - 49, 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 - 50*x^11 + 25*x^10 + 722*x^9 - 226*x^8 - 4207*x^7 + 1158*x^6 + 10465*x^5 - 2535*x^4 - 9399*x^3 + 1079*x^2 + 1316*x - 49);
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 - 50*x^11 + 25*x^10 + 722*x^9 - 226*x^8 - 4207*x^7 + 1158*x^6 + 10465*x^5 - 2535*x^4 - 9399*x^3 + 1079*x^2 + 1316*x - 49);
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 26 |
| The 8 conjugacy class representatives for $D_{13}$ |
| Character table for $D_{13}$ |
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
| Galois closure: | deg 26 |
| 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 | ${\href{/padicField/2.13.0.1}{13} }$ | ${\href{/padicField/3.2.0.1}{2} }^{6}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.13.0.1}{13} }$ | ${\href{/padicField/7.2.0.1}{2} }^{6}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.13.0.1}{13} }$ | ${\href{/padicField/13.13.0.1}{13} }$ | ${\href{/padicField/17.13.0.1}{13} }$ | ${\href{/padicField/19.13.0.1}{13} }$ | R | ${\href{/padicField/29.2.0.1}{2} }^{6}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.13.0.1}{13} }$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.13.0.1}{13} }$ | ${\href{/padicField/43.13.0.1}{13} }$ | ${\href{/padicField/47.13.0.1}{13} }$ | ${\href{/padicField/53.2.0.1}{2} }^{6}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.13.0.1}{13} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(23\)
| $\Q_{23}$ | $x + 18$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 23$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
|
\(367\)
| $\Q_{367}$ | $x$ | $1$ | $1$ | $0$ | 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 $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |