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Properties

Label 15.1.35347664861...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{23}\cdot 5^{10}\cdot 19^{13}\cdot 29^{13}$
Root discriminant $2010.13$
Ramified primes $2, 5, 19, 29$
Class number $250$ (GRH)
Class group $[5, 5, 10]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-294690202775308898, 57762542818688054, -5829949369516366, 481533826058428, -47700853439788, 2607605182191, 68139475134, 4164671679, -945266382, -22740148, 1678622, 148616, 956, -623, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 2*x^14 - 623*x^13 + 956*x^12 + 148616*x^11 + 1678622*x^10 - 22740148*x^9 - 945266382*x^8 + 4164671679*x^7 + 68139475134*x^6 + 2607605182191*x^5 - 47700853439788*x^4 + 481533826058428*x^3 - 5829949369516366*x^2 + 57762542818688054*x - 294690202775308898)
 
gp: K = bnfinit(x^15 - 2*x^14 - 623*x^13 + 956*x^12 + 148616*x^11 + 1678622*x^10 - 22740148*x^9 - 945266382*x^8 + 4164671679*x^7 + 68139475134*x^6 + 2607605182191*x^5 - 47700853439788*x^4 + 481533826058428*x^3 - 5829949369516366*x^2 + 57762542818688054*x - 294690202775308898, 1)
 

Normalized defining polynomial

\( x^{15} - 2 x^{14} - 623 x^{13} + 956 x^{12} + 148616 x^{11} + 1678622 x^{10} - 22740148 x^{9} - 945266382 x^{8} + 4164671679 x^{7} + 68139475134 x^{6} + 2607605182191 x^{5} - 47700853439788 x^{4} + 481533826058428 x^{3} - 5829949369516366 x^{2} + 57762542818688054 x - 294690202775308898 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-35347664861463185809488152464433328209920000000000=-\,2^{23}\cdot 5^{10}\cdot 19^{13}\cdot 29^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2010.13$
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:  $2, 5, 19, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{55} a^{6} - \frac{4}{55} a^{5} + \frac{18}{55} a^{4} - \frac{16}{55} a^{3} + \frac{2}{55} a^{2} + \frac{18}{55} a + \frac{5}{11}$, $\frac{1}{55} a^{7} + \frac{2}{55} a^{5} + \frac{1}{55} a^{4} - \frac{7}{55} a^{3} + \frac{26}{55} a^{2} - \frac{13}{55} a - \frac{2}{11}$, $\frac{1}{1045} a^{8} - \frac{9}{1045} a^{7} + \frac{6}{1045} a^{6} + \frac{4}{95} a^{5} - \frac{32}{1045} a^{4} + \frac{476}{1045} a^{3} - \frac{371}{1045} a^{2} - \frac{492}{1045} a + \frac{421}{1045}$, $\frac{1}{11495} a^{9} + \frac{2}{11495} a^{8} + \frac{78}{11495} a^{7} - \frac{4}{11495} a^{6} - \frac{631}{11495} a^{5} + \frac{1587}{11495} a^{4} + \frac{5074}{11495} a^{3} + \frac{3616}{11495} a^{2} - \frac{1533}{11495} a - \frac{1392}{11495}$, $\frac{1}{57475} a^{10} + \frac{2}{57475} a^{9} + \frac{12}{57475} a^{8} - \frac{246}{57475} a^{7} + \frac{436}{57475} a^{6} + \frac{2654}{57475} a^{5} - \frac{24791}{57475} a^{4} + \frac{12119}{57475} a^{3} + \frac{27133}{57475} a^{2} - \frac{12183}{57475} a + \frac{1559}{5225}$, $\frac{1}{57475} a^{11} - \frac{2}{57475} a^{9} - \frac{3}{11495} a^{8} - \frac{237}{57475} a^{7} + \frac{337}{57475} a^{6} + \frac{5031}{57475} a^{5} - \frac{911}{3025} a^{4} + \frac{727}{11495} a^{3} + \frac{15861}{57475} a^{2} + \frac{2074}{11495} a - \frac{3878}{57475}$, $\frac{1}{632225} a^{12} + \frac{3}{632225} a^{11} + \frac{1}{632225} a^{10} - \frac{3}{126445} a^{9} - \frac{136}{632225} a^{8} - \frac{3147}{632225} a^{7} + \frac{766}{126445} a^{6} + \frac{59701}{632225} a^{5} + \frac{23878}{126445} a^{4} + \frac{293133}{632225} a^{3} + \frac{315902}{632225} a^{2} - \frac{20777}{57475} a + \frac{38053}{632225}$, $\frac{1}{6954475} a^{13} + \frac{1}{1390895} a^{12} + \frac{7}{6954475} a^{11} - \frac{2}{6954475} a^{10} - \frac{144}{6954475} a^{9} - \frac{2077}{6954475} a^{8} + \frac{67}{25289} a^{7} + \frac{21942}{6954475} a^{6} + \frac{114316}{6954475} a^{5} - \frac{1285353}{6954475} a^{4} - \frac{2388823}{6954475} a^{3} + \frac{36321}{278179} a^{2} - \frac{1113889}{6954475} a - \frac{19923}{73205}$, $\frac{1}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a^{14} - \frac{252196853046457208443508762265040894479595994029187893938334546868}{264050769175057051240694043510005202409240605737489161778352007205609635045} a^{13} + \frac{11099296004495549118534847469504434083843518509101036119828831167675374}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a^{12} - \frac{40822396742401469231465691570992917542516876324845872095985012195158018}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a^{11} + \frac{51969375429384642513704503533019390261493800150282654746143248373619691}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a^{10} - \frac{32474004442857102452476312416973816276631819754998923631319599736718982}{1560299999670791666422282984377303468781876306630617774144807315305875116175} a^{9} + \frac{363421144937387564430786506881002771049117614790285696439406523611356912}{2451899999482672618663587546878619736657234196133827930798982924052089468275} a^{8} - \frac{130531228415205487686945290320109567925429174701830578870509221704704124116}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a^{7} - \frac{29841557722988429664280941286886986548376848687003986329522843022264537826}{3432659999275741666129022565630067631320127874587359103118576093672925255585} a^{6} + \frac{15996626565445248310516370302922350355957094456629406003597086081810384893}{1320253845875285256203470217550026012046203028687445808891760036028048175225} a^{5} - \frac{642636053686745047555088439336897465501249706978045617449720340014434617288}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a^{4} - \frac{6789012220970849710295607715180808669773340120399453261055708227190454163807}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a^{3} - \frac{5743180211217260970992821332450343423292811269795230274211968666585592825788}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a^{2} + \frac{919470600645494461349587641054104641598341210722852899964161929002614849422}{17163299996378708330645112828150338156600639372936795515592880468364626277925} a - \frac{2944282263813171742803623636558508613739514832243293432412150401817780248471}{17163299996378708330645112828150338156600639372936795515592880468364626277925}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{10}$, which has order $250$ (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:  $7$
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:  \( 447144440989138750 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_5$ (as 15T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 120
The 15 conjugacy class representatives for $F_5 \times S_3$
Character table for $F_5 \times S_3$

Intermediate fields

3.1.22040.1, 5.1.184347134402000.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{15}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.10.9.2$x^{10} + 76$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$
$29$29.5.4.1$x^{5} - 29$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
29.10.9.1$x^{10} - 29$$10$$1$$9$$D_{10}$$[\ ]_{10}^{2}$