↑

Properties

Label 15.11.2503040973...0000.1
Degree $15$
Signature $[11, 2]$
Discriminant $2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 13^{2}\cdot 421\cdot 66361^{4}\cdot 3959620477^{2}$
Root discriminant $12{,}394.56$
Ramified primes $2, 5, 7, 13, 421, 66361, 3959620477$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 15T101

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-139169103872000, -365318897664000, -413593180569600, -263497127526400, -103333059624960, -25637180200960, -3989611359232, -376447428736, -20997415296, -818687880, -36325376, -1149208, -3744, -234, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 234*x^13 - 3744*x^12 - 1149208*x^11 - 36325376*x^10 - 818687880*x^9 - 20997415296*x^8 - 376447428736*x^7 - 3989611359232*x^6 - 25637180200960*x^5 - 103333059624960*x^4 - 263497127526400*x^3 - 413593180569600*x^2 - 365318897664000*x - 139169103872000)
 
gp: K = bnfinit(x^15 - 234*x^13 - 3744*x^12 - 1149208*x^11 - 36325376*x^10 - 818687880*x^9 - 20997415296*x^8 - 376447428736*x^7 - 3989611359232*x^6 - 25637180200960*x^5 - 103333059624960*x^4 - 263497127526400*x^3 - 413593180569600*x^2 - 365318897664000*x - 139169103872000, 1)
 

Normalized defining polynomial

\( x^{15} - 234 x^{13} - 3744 x^{12} - 1149208 x^{11} - 36325376 x^{10} - 818687880 x^{9} - 20997415296 x^{8} - 376447428736 x^{7} - 3989611359232 x^{6} - 25637180200960 x^{5} - 103333059624960 x^{4} - 263497127526400 x^{3} - 413593180569600 x^{2} - 365318897664000 x - 139169103872000 \)

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

Invariants

Degree:  $15$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(25030409738237508654535690145991819085872813409237356544000000=2^{18}\cdot 5^{6}\cdot 7^{10}\cdot 13^{2}\cdot 421\cdot 66361^{4}\cdot 3959620477^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12{,}394.56$
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:  $2, 5, 7, 13, 421, 66361, 3959620477$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{20} a^{5} + \frac{1}{10} a^{3} + \frac{1}{10} a^{2}$, $\frac{1}{40} a^{6} + \frac{1}{20} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{160} a^{7} - \frac{1}{80} a^{5} - \frac{1}{20} a^{4} - \frac{1}{20} a^{3} + \frac{1}{5} a^{2} - \frac{1}{4} a$, $\frac{1}{3200} a^{8} - \frac{13}{1600} a^{6} + \frac{1}{50} a^{5} - \frac{47}{400} a^{4} - \frac{4}{25} a^{3} + \frac{23}{400} a^{2} + \frac{1}{5}$, $\frac{1}{102400} a^{9} + \frac{27}{51200} a^{7} - \frac{13}{3200} a^{6} + \frac{213}{12800} a^{5} - \frac{1}{200} a^{4} - \frac{17}{12800} a^{3} - \frac{39}{160} a^{2} + \frac{41}{160} a - \frac{1}{4}$, $\frac{1}{409600} a^{10} + \frac{27}{204800} a^{8} - \frac{13}{12800} a^{7} - \frac{427}{51200} a^{6} + \frac{9}{800} a^{5} - \frac{1297}{51200} a^{4} + \frac{41}{640} a^{3} + \frac{137}{640} a^{2} + \frac{7}{16} a$, $\frac{1}{65536000} a^{11} - \frac{1}{1638400} a^{10} + \frac{83}{32768000} a^{9} + \frac{511}{4096000} a^{8} + \frac{2409}{8192000} a^{7} + \frac{9271}{1024000} a^{6} + \frac{32467}{1638400} a^{5} - \frac{77649}{1024000} a^{4} + \frac{17559}{256000} a^{3} + \frac{547}{4000} a^{2} + \frac{1427}{6400} a - \frac{71}{800}$, $\frac{1}{1310720000} a^{12} - \frac{77}{655360000} a^{10} - \frac{17}{40960000} a^{9} + \frac{21209}{163840000} a^{8} + \frac{6409}{5120000} a^{7} - \frac{392813}{32768000} a^{6} - \frac{16667}{10240000} a^{5} - \frac{342331}{5120000} a^{4} - \frac{68969}{640000} a^{3} - \frac{28381}{128000} a^{2} + \frac{409}{1000} a + \frac{81}{400}$, $\frac{1}{2621440000000} a^{13} - \frac{3}{16384000000} a^{12} + \frac{3403}{1310720000000} a^{11} + \frac{23193}{81920000000} a^{10} + \frac{902269}{327680000000} a^{9} - \frac{843241}{10240000000} a^{8} + \frac{140768387}{65536000000} a^{7} - \frac{77930837}{20480000000} a^{6} + \frac{399793623}{20480000000} a^{5} + \frac{140453067}{2560000000} a^{4} + \frac{8423989}{256000000} a^{3} + \frac{1490677}{8000000} a^{2} + \frac{1005961}{3200000} a - \frac{71457}{400000}$, $\frac{1}{2684354560000000} a^{14} - \frac{43}{335544320000000} a^{13} - \frac{465237}{1342177280000000} a^{12} + \frac{774237}{167772160000000} a^{11} - \frac{196094739}{335544320000000} a^{10} - \frac{138417391}{41943040000000} a^{9} - \frac{17787550897}{335544320000000} a^{8} - \frac{52385880779}{41943040000000} a^{7} - \frac{18863097709}{20971520000000} a^{6} - \frac{7757499023}{655360000000} a^{5} - \frac{101548007499}{1310720000000} a^{4} + \frac{7360241221}{32768000000} a^{3} - \frac{3510557551}{16384000000} a^{2} + \frac{15144253}{40960000} a + \frac{5981481}{51200000}$

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:  $12$
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:  \( 68524043832000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

15T101:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5184000
The 133 conjugacy class representatives for [S(5)^3]3=S(5)wr3 are not computed
Character table for [S(5)^3]3=S(5)wr3 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 18 sibling: data not computed
Degree 30 siblings: data not computed
Degree 36 siblings: data not computed
Degree 45 sibling: 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 R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R $15$ ${\href{/LocalNumberField/19.9.0.1}{9} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$5$5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
421Data not computed
66361Data not computed
3959620477Data not computed