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Properties

Label 15.9.31256653180...1875.1
Degree $15$
Signature $[9, 3]$
Discriminant $-\,5^{5}\cdot 13^{3}\cdot 19^{6}\cdot 31^{10}\cdot 41^{4}\cdot 347^{3}$
Root discriminant $793.71$
Ramified primes $5, 13, 19, 31, 41, 347$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 15T70

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-414601786688651713, 414276426535790729, 33906382851120779, -19327557996736266, -75831436048429, 60691267440950, 2695173632356, -106782015292, -11155818269, 207980393, 14315825, 194086, -3539, -1264, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 4*x^14 - 1264*x^13 - 3539*x^12 + 194086*x^11 + 14315825*x^10 + 207980393*x^9 - 11155818269*x^8 - 106782015292*x^7 + 2695173632356*x^6 + 60691267440950*x^5 - 75831436048429*x^4 - 19327557996736266*x^3 + 33906382851120779*x^2 + 414276426535790729*x - 414601786688651713)
 
gp: K = bnfinit(x^15 - 4*x^14 - 1264*x^13 - 3539*x^12 + 194086*x^11 + 14315825*x^10 + 207980393*x^9 - 11155818269*x^8 - 106782015292*x^7 + 2695173632356*x^6 + 60691267440950*x^5 - 75831436048429*x^4 - 19327557996736266*x^3 + 33906382851120779*x^2 + 414276426535790729*x - 414601786688651713, 1)
 

Normalized defining polynomial

\( x^{15} - 4 x^{14} - 1264 x^{13} - 3539 x^{12} + 194086 x^{11} + 14315825 x^{10} + 207980393 x^{9} - 11155818269 x^{8} - 106782015292 x^{7} + 2695173632356 x^{6} + 60691267440950 x^{5} - 75831436048429 x^{4} - 19327557996736266 x^{3} + 33906382851120779 x^{2} + 414276426535790729 x - 414601786688651713 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-31256653180970842100294052450870850335221875=-\,5^{5}\cdot 13^{3}\cdot 19^{6}\cdot 31^{10}\cdot 41^{4}\cdot 347^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $793.71$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $5, 13, 19, 31, 41, 347$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{14} - \frac{2071951256415232704139612875243018329994290272360451595508583995478405460987793350861787310049900894774547412121484951745}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{13} + \frac{1532322737080521567766954543131929092521285541263605682574062740989466711233063739331313422227160886629429662293005013355}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{12} + \frac{3703452571988009293850794702475621585620395108398010489361827437107187117895235635643995880090514946642689457905793055638}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{11} + \frac{797813418878962912563216754452791964728252196528806731545835672297766005635001773970837561161592700348571306342700197990}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{10} + \frac{74693151246234852542679639969774273010863999290687148367921900926408147987950740971370214705343128373176189150453145687}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{9} + \frac{3772888278446641831534661628885514241666635984996904931891065720426514339185533099105125258412072681010492198346229523107}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{8} + \frac{1735329755267165606177204616027881578451795050164790276043651795024122607839930157512643791253885701201291580378420705700}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{7} + \frac{967882621723417657642693171523147492226346414419950207602237490951332087469814820924520507427144687238914795291993737590}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{6} - \frac{3519476344139011541989340800450147412643529685870041481979454060769139066303897650295719387727539911703763463884226220812}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{5} + \frac{2779243267532072740593603282235869840767859174871757416496607606032489968631173334065456753980374009464041462033608469667}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{4} + \frac{3689997612082150133153303608317282210868256890940791389401937985600669944815136672169317778200538628646205614323696220479}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{3} - \frac{2385191592487872657366543856418735391488177921172173921274659327752451687328142669203303397446984123342100478218078868850}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a^{2} + \frac{309618247603472435728233759833862336217867120772944434097403980902493023091004002783650827175239887630075393293566094349}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539} a - \frac{965830446367180324527906705250463393878528122186528178683329388886907760383357551476537791560002710513741337442002657899}{9828223129945599856007751018231218548002242151397666087625749494601254997512335959894810029148192233088975448764411838539}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 8810665161660000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T70:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 19440
The 36 conjugacy class representatives for [3^4:2]S(5)
Character table for [3^4:2]S(5) is not computed

Intermediate fields

5.3.4511.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 30 siblings: data not computed
Degree 45 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ R ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ R ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.5.0.1$x^{5} - x + 2$$1$$5$$0$$C_5$$[\ ]^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.9.6.3$x^{9} - 361 x^{3} + 27436$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
$31$31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.12.8.3$x^{12} + 32674 x^{6} - 119164 x^{3} + 266897569$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.6.4.2$x^{6} - 41 x^{3} + 20172$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$
347Data not computed