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Properties

Label 15.15.2101141399...0000.1
Degree $15$
Signature $[15, 0]$
Discriminant $2^{10}\cdot 5^{10}\cdot 19^{12}\cdot 37^{7}$
Root discriminant $263.94$
Ramified primes $2, 5, 19, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T40

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![981137, -18776851, -35952686, 42035037, 73214629, 139474, -25199016, -2555513, 3284150, 134362, -198002, 11225, 3108, -297, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 7*x^14 - 297*x^13 + 3108*x^12 + 11225*x^11 - 198002*x^10 + 134362*x^9 + 3284150*x^8 - 2555513*x^7 - 25199016*x^6 + 139474*x^5 + 73214629*x^4 + 42035037*x^3 - 35952686*x^2 - 18776851*x + 981137)
 
gp: K = bnfinit(x^15 - 7*x^14 - 297*x^13 + 3108*x^12 + 11225*x^11 - 198002*x^10 + 134362*x^9 + 3284150*x^8 - 2555513*x^7 - 25199016*x^6 + 139474*x^5 + 73214629*x^4 + 42035037*x^3 - 35952686*x^2 - 18776851*x + 981137, 1)
 

Normalized defining polynomial

\( x^{15} - 7 x^{14} - 297 x^{13} + 3108 x^{12} + 11225 x^{11} - 198002 x^{10} + 134362 x^{9} + 3284150 x^{8} - 2555513 x^{7} - 25199016 x^{6} + 139474 x^{5} + 73214629 x^{4} + 42035037 x^{3} - 35952686 x^{2} - 18776851 x + 981137 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[15, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2101141399534246351499964130000000000=2^{10}\cdot 5^{10}\cdot 19^{12}\cdot 37^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $263.94$
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:  $2, 5, 19, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{8} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{125} a^{13} + \frac{11}{125} a^{12} - \frac{3}{125} a^{10} - \frac{4}{125} a^{9} + \frac{54}{125} a^{8} + \frac{38}{125} a^{7} + \frac{1}{25} a^{6} - \frac{61}{125} a^{5} - \frac{44}{125} a^{4} + \frac{18}{125} a^{3} - \frac{53}{125} a^{2} - \frac{12}{25} a + \frac{37}{125}$, $\frac{1}{5895690897938061455971224355818665740443045060125} a^{14} + \frac{3547621791576494935313306648758136723956016729}{1179138179587612291194244871163733148088609012025} a^{13} + \frac{305028768348923531062522997788505976374886786599}{5895690897938061455971224355818665740443045060125} a^{12} + \frac{414630450924128387324900675095983675004009639522}{5895690897938061455971224355818665740443045060125} a^{11} - \frac{152340357125005637894652943734979139212956478331}{5895690897938061455971224355818665740443045060125} a^{10} + \frac{479734556205814159668802950464238356256497194143}{5895690897938061455971224355818665740443045060125} a^{9} + \frac{1184559965854789812333430443661352459094849830674}{5895690897938061455971224355818665740443045060125} a^{8} - \frac{2279347866024861046926627730818773370372662795153}{5895690897938061455971224355818665740443045060125} a^{7} + \frac{2024150004645787844341336022116456411476800398684}{5895690897938061455971224355818665740443045060125} a^{6} + \frac{2484835296856501206553825359208165630582389875657}{5895690897938061455971224355818665740443045060125} a^{5} - \frac{2501398388426795032936111161528602596096518524003}{5895690897938061455971224355818665740443045060125} a^{4} - \frac{682662203746574536122605739485066247503821291616}{5895690897938061455971224355818665740443045060125} a^{3} - \frac{29590211505810629329275483177343256678882360727}{190183577352840692128104011478021475498162743875} a^{2} + \frac{2533583729413870714211216800470345982396077186922}{5895690897938061455971224355818665740443045060125} a - \frac{2687812551113183692131557599569680882922413787742}{5895690897938061455971224355818665740443045060125}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 21270551246000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T40:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1500
The 40 conjugacy class representatives for [5^3:2]S(3)
Character table for [5^3:2]S(3) is not computed

Intermediate fields

3.3.148.1

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

Sibling fields

Degree 15 siblings: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $15$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ R ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R $15$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.5.6.1$x^{5} + 10 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
$19$19.5.4.1$x^{5} - 19$$5$$1$$4$$D_{5}$$[\ ]_{5}^{2}$
19.10.8.2$x^{10} - 19 x^{5} + 722$$5$$2$$8$$D_5\times C_5$$[\ ]_{5}^{10}$
37Data not computed