↑

Properties

Label 15.1.81768020500...0000.1
Degree $15$
Signature $[1, 7]$
Discriminant $-\,2^{12}\cdot 5^{15}\cdot 11^{13}\cdot 1801^{5}$
Root discriminant $846.26$
Ramified primes $2, 5, 11, 1801$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $F_5 \times S_3$ (as 15T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![350125148, 280653460, -2612120, 37747640, 7024160, 552212, 422300, 560000, -9120, 10770, 6533, -495, 130, 40, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 5*x^14 + 40*x^13 + 130*x^12 - 495*x^11 + 6533*x^10 + 10770*x^9 - 9120*x^8 + 560000*x^7 + 422300*x^6 + 552212*x^5 + 7024160*x^4 + 37747640*x^3 - 2612120*x^2 + 280653460*x + 350125148)
 
gp: K = bnfinit(x^15 - 5*x^14 + 40*x^13 + 130*x^12 - 495*x^11 + 6533*x^10 + 10770*x^9 - 9120*x^8 + 560000*x^7 + 422300*x^6 + 552212*x^5 + 7024160*x^4 + 37747640*x^3 - 2612120*x^2 + 280653460*x + 350125148, 1)
 

Normalized defining polynomial

\( x^{15} - 5 x^{14} + 40 x^{13} + 130 x^{12} - 495 x^{11} + 6533 x^{10} + 10770 x^{9} - 9120 x^{8} + 560000 x^{7} + 422300 x^{6} + 552212 x^{5} + 7024160 x^{4} + 37747640 x^{3} - 2612120 x^{2} + 280653460 x + 350125148 \)

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:  \(-81768020500282951857411490366375000000000000=-\,2^{12}\cdot 5^{15}\cdot 11^{13}\cdot 1801^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $846.26$
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:  $2, 5, 11, 1801$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{22} a^{10} + \frac{1}{22} a^{9} - \frac{4}{11} a^{8} - \frac{2}{11} a^{7} - \frac{3}{22} a^{6} + \frac{7}{22} a^{5} - \frac{1}{11} a^{4} + \frac{4}{11} a^{3} - \frac{2}{11} a^{2} + \frac{2}{11} a + \frac{5}{11}$, $\frac{1}{44} a^{11} - \frac{1}{44} a^{10} + \frac{3}{11} a^{9} - \frac{5}{22} a^{8} + \frac{5}{44} a^{7} - \frac{9}{44} a^{6} - \frac{4}{11} a^{5} + \frac{3}{11} a^{4} - \frac{5}{11} a^{3} - \frac{5}{22} a^{2} + \frac{1}{22} a - \frac{5}{11}$, $\frac{1}{44} a^{12} - \frac{1}{44} a^{10} - \frac{5}{22} a^{9} + \frac{3}{44} a^{8} + \frac{1}{4} a^{6} + \frac{4}{11} a^{4} + \frac{3}{22} a^{3} - \frac{1}{11} a^{2} - \frac{1}{2} a - \frac{2}{11}$, $\frac{1}{44} a^{13} - \frac{1}{44} a^{10} - \frac{19}{44} a^{9} - \frac{1}{22} a^{8} + \frac{5}{11} a^{7} + \frac{5}{44} a^{6} - \frac{9}{22} a^{5} - \frac{1}{22} a^{4} + \frac{3}{11} a^{3} + \frac{4}{11} a^{2} - \frac{5}{22} a - \frac{2}{11}$, $\frac{1}{20315682583372749850806533823979203653399505364} a^{14} - \frac{15707184793701329848374343949822194857028304}{5078920645843187462701633455994800913349876341} a^{13} - \frac{61574351834629260639586377512518070429873693}{10157841291686374925403266911989601826699752682} a^{12} - \frac{117539600449448516928874994693431235911532601}{20315682583372749850806533823979203653399505364} a^{11} - \frac{74173665347774785765360596634983999251464871}{20315682583372749850806533823979203653399505364} a^{10} + \frac{2279360733857892831345950282579094600222395832}{5078920645843187462701633455994800913349876341} a^{9} - \frac{1701422585307883399411120530798959235310179213}{10157841291686374925403266911989601826699752682} a^{8} + \frac{6634175881268993837313971821303311513769171077}{20315682583372749850806533823979203653399505364} a^{7} + \frac{1616156420516363396172377942097919755481392409}{10157841291686374925403266911989601826699752682} a^{6} - \frac{846772164355219559883786209350616826370180174}{5078920645843187462701633455994800913349876341} a^{5} - \frac{550302013912758998615847528223451208025234380}{5078920645843187462701633455994800913349876341} a^{4} - \frac{113999441513899286628900534058914742045016971}{5078920645843187462701633455994800913349876341} a^{3} - \frac{3902395105259363241350481933877301424493706233}{10157841291686374925403266911989601826699752682} a^{2} + \frac{30074021647197552142486255401784443684401182}{461720058713017042063784859635890992122716031} a - \frac{1796395619060167428515261648447806726602021675}{5078920645843187462701633455994800913349876341}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 799236990706202.1 \) (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

3.1.19811.1, 5.1.732050000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.15.15.40$x^{15} + 10 x^{14} + 5 x^{13} + 20 x^{12} + 20 x^{11} + 17 x^{10} + 10 x^{9} + 15 x^{8} + 12 x^{5} + 15 x^{3} + 10 x^{2} + 20 x + 17$$5$$3$$15$$F_5\times C_3$$[5/4]_{4}^{3}$
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.9.4$x^{10} - 99$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1801Data not computed