Properties

Label 25.5.396...449.1
Degree $25$
Signature $[5, 10]$
Discriminant $3.967\times 10^{34}$
Root discriminant $24.21$
Ramified primes $3, 31$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_5\times D_5$ (as 25T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^25 - 3*x^24 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*x - 1)
 
gp: K = bnfinit(x^25 - 3*x^24 + x^23 - 3*x^22 + 19*x^21 - 17*x^20 - 8*x^19 - 27*x^18 + 13*x^17 + 72*x^16 + 7*x^15 + 58*x^14 - 37*x^13 - 186*x^12 + 151*x^11 - 314*x^10 + 404*x^9 - 330*x^8 + 297*x^7 - 201*x^6 + 123*x^5 - 65*x^4 + 32*x^3 - 18*x^2 + 7*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 7, -18, 32, -65, 123, -201, 297, -330, 404, -314, 151, -186, -37, 58, 7, 72, 13, -27, -8, -17, 19, -3, 1, -3, 1]);
 

\(x^{25} - 3 x^{24} + x^{23} - 3 x^{22} + 19 x^{21} - 17 x^{20} - 8 x^{19} - 27 x^{18} + 13 x^{17} + 72 x^{16} + 7 x^{15} + 58 x^{14} - 37 x^{13} - 186 x^{12} + 151 x^{11} - 314 x^{10} + 404 x^{9} - 330 x^{8} + 297 x^{7} - 201 x^{6} + 123 x^{5} - 65 x^{4} + 32 x^{3} - 18 x^{2} + 7 x - 1\)  Toggle raw display

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

Invariants

Degree:  $25$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[5, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(39668558936237989671952184481937449\)\(\medspace = 3^{10}\cdot 31^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $24.21$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 31$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $5$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} + \frac{2}{5} a^{17} - \frac{2}{5} a^{14} - \frac{2}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{10} + \frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{17} - \frac{2}{5} a^{15} + \frac{2}{5} a^{14} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{20} - \frac{2}{5} a^{17} - \frac{2}{5} a^{16} + \frac{2}{5} a^{15} - \frac{2}{5} a^{14} - \frac{2}{5} a^{13} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{21} + \frac{2}{5} a^{17} + \frac{2}{5} a^{16} - \frac{2}{5} a^{15} - \frac{1}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{22} - \frac{2}{5} a^{17} - \frac{2}{5} a^{16} - \frac{1}{5} a^{15} - \frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{23} + \frac{2}{5} a^{17} - \frac{1}{5} a^{16} + \frac{1}{5} a^{13} + \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{1}{5} a^{9} + \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5}$, $\frac{1}{17632972971007882844275} a^{24} + \frac{630527257748199744148}{17632972971007882844275} a^{23} - \frac{1503136271450097471461}{17632972971007882844275} a^{22} + \frac{41270282286519764116}{17632972971007882844275} a^{21} + \frac{5886160851694034599}{705318918840315313771} a^{20} + \frac{966420409645934781843}{17632972971007882844275} a^{19} - \frac{293833042744993878443}{3526594594201576568855} a^{18} - \frac{2585480418227484639582}{17632972971007882844275} a^{17} + \frac{614296213257403297781}{17632972971007882844275} a^{16} + \frac{7769022460209152823263}{17632972971007882844275} a^{15} + \frac{1643358802436313537061}{3526594594201576568855} a^{14} - \frac{3566784763985250222167}{17632972971007882844275} a^{13} - \frac{6414991158414955301169}{17632972971007882844275} a^{12} - \frac{19385717330246834124}{3526594594201576568855} a^{11} + \frac{754725455249841113096}{17632972971007882844275} a^{10} - \frac{749577747651540666533}{17632972971007882844275} a^{9} + \frac{5272979305765968710761}{17632972971007882844275} a^{8} + \frac{8170342893873747989356}{17632972971007882844275} a^{7} - \frac{6407436097885275451047}{17632972971007882844275} a^{6} - \frac{4475000120988456821518}{17632972971007882844275} a^{5} - \frac{386912676715001167184}{3526594594201576568855} a^{4} + \frac{418722577707623519437}{3526594594201576568855} a^{3} + \frac{1829266319254630547827}{17632972971007882844275} a^{2} + \frac{5120635846035271204559}{17632972971007882844275} a - \frac{4779019204138182365199}{17632972971007882844275}$  Toggle raw display

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $14$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 68852831.13170123 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{10}\cdot 68852831.13170123 \cdot 1}{2\sqrt{39668558936237989671952184481937449}}\approx 0.530416782562469$ (assuming GRH)

Galois group

$C_5\times D_5$ (as 25T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 50
The 20 conjugacy class representatives for $C_5\times D_5$
Character table for $C_5\times D_5$

Intermediate fields

5.5.923521.1, 5.1.8311689.1

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ R ${\href{/LocalNumberField/37.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$3$3.5.0.1$x^{5} - x + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
31Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.93.10t1.a.c$1$ $ 3 \cdot 31 $ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.31.5t1.a.d$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
* 1.31.5t1.a.b$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
1.93.10t1.a.d$1$ $ 3 \cdot 31 $ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.93.10t1.a.b$1$ $ 3 \cdot 31 $ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
1.93.10t1.a.a$1$ $ 3 \cdot 31 $ 10.0.207252522098163.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.31.5t1.a.a$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
* 1.31.5t1.a.c$1$ $ 31 $ 5.5.923521.1 $C_5$ (as 5T1) $0$ $1$
* 2.93.10t6.a.a$2$ $ 3 \cdot 31 $ 25.5.39668558936237989671952184481937449.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2883.10t6.a.b$2$ $ 3 \cdot 31^{2}$ 25.5.39668558936237989671952184481937449.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2883.5t2.a.a$2$ $ 3 \cdot 31^{2}$ 5.1.8311689.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.2883.10t6.a.a$2$ $ 3 \cdot 31^{2}$ 25.5.39668558936237989671952184481937449.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2883.10t6.a.c$2$ $ 3 \cdot 31^{2}$ 25.5.39668558936237989671952184481937449.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2883.10t6.a.d$2$ $ 3 \cdot 31^{2}$ 25.5.39668558936237989671952184481937449.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.93.10t6.a.c$2$ $ 3 \cdot 31 $ 25.5.39668558936237989671952184481937449.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.93.10t6.a.b$2$ $ 3 \cdot 31 $ 25.5.39668558936237989671952184481937449.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.2883.5t2.a.b$2$ $ 3 \cdot 31^{2}$ 5.1.8311689.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.93.10t6.a.d$2$ $ 3 \cdot 31 $ 25.5.39668558936237989671952184481937449.1 $C_5\times D_5$ (as 25T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.