Properties

Label 25.5.722...624.1
Degree $25$
Signature $[5, 10]$
Discriminant $7.224\times 10^{29}$
Root discriminant $15.64$
Ramified primes $2, 11$
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 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1)
 
gp: K = bnfinit(x^25 - 3*x^24 + 4*x^23 - 7*x^22 + 5*x^21 - 6*x^20 + 28*x^19 - 37*x^18 + 24*x^17 + 60*x^15 - 133*x^14 + 74*x^13 + 37*x^12 - 24*x^11 - 112*x^10 + 155*x^9 - 56*x^8 - 45*x^7 + 51*x^6 - 17*x^5 + 3*x^4 - 3*x^3 + 3*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 3, 0, -3, 3, -17, 51, -45, -56, 155, -112, -24, 37, 74, -133, 60, 0, 24, -37, 28, -6, 5, -7, 4, -3, 1]);
 

\( x^{25} - 3 x^{24} + 4 x^{23} - 7 x^{22} + 5 x^{21} - 6 x^{20} + 28 x^{19} - 37 x^{18} + 24 x^{17} + 60 x^{15} - 133 x^{14} + 74 x^{13} + 37 x^{12} - 24 x^{11} - 112 x^{10} + 155 x^{9} - 56 x^{8} - 45 x^{7} + 51 x^{6} - 17 x^{5} + 3 x^{4} - 3 x^{3} + 3 x - 1 \)

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:  \(722359806654877741268938522624\)\(\medspace = 2^{30}\cdot 11^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.64$
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:  $2, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{19} a^{20} + \frac{6}{19} a^{19} - \frac{9}{19} a^{18} - \frac{2}{19} a^{17} + \frac{5}{19} a^{16} + \frac{5}{19} a^{15} + \frac{9}{19} a^{13} + \frac{2}{19} a^{12} + \frac{3}{19} a^{11} + \frac{2}{19} a^{10} - \frac{6}{19} a^{9} + \frac{8}{19} a^{8} - \frac{1}{19} a^{7} + \frac{4}{19} a^{6} - \frac{5}{19} a^{5} - \frac{5}{19} a^{4} + \frac{8}{19} a^{3} + \frac{7}{19} a^{2} - \frac{8}{19} a + \frac{3}{19}$, $\frac{1}{19} a^{21} - \frac{7}{19} a^{19} - \frac{5}{19} a^{18} - \frac{2}{19} a^{17} - \frac{6}{19} a^{16} + \frac{8}{19} a^{15} + \frac{9}{19} a^{14} + \frac{5}{19} a^{13} - \frac{9}{19} a^{12} + \frac{3}{19} a^{11} + \frac{1}{19} a^{10} + \frac{6}{19} a^{9} + \frac{8}{19} a^{8} - \frac{9}{19} a^{7} + \frac{9}{19} a^{6} + \frac{6}{19} a^{5} - \frac{3}{19} a^{3} + \frac{7}{19} a^{2} - \frac{6}{19} a + \frac{1}{19}$, $\frac{1}{19} a^{22} - \frac{1}{19} a^{19} - \frac{8}{19} a^{18} - \frac{1}{19} a^{17} + \frac{5}{19} a^{16} + \frac{6}{19} a^{15} + \frac{5}{19} a^{14} - \frac{3}{19} a^{13} - \frac{2}{19} a^{12} + \frac{3}{19} a^{11} + \frac{1}{19} a^{10} + \frac{4}{19} a^{9} + \frac{9}{19} a^{8} + \frac{2}{19} a^{7} - \frac{4}{19} a^{6} + \frac{3}{19} a^{5} + \frac{6}{19} a^{3} + \frac{5}{19} a^{2} + \frac{2}{19} a + \frac{2}{19}$, $\frac{1}{19} a^{23} - \frac{2}{19} a^{19} + \frac{9}{19} a^{18} + \frac{3}{19} a^{17} - \frac{8}{19} a^{16} - \frac{9}{19} a^{15} - \frac{3}{19} a^{14} + \frac{7}{19} a^{13} + \frac{5}{19} a^{12} + \frac{4}{19} a^{11} + \frac{6}{19} a^{10} + \frac{3}{19} a^{9} - \frac{9}{19} a^{8} - \frac{5}{19} a^{7} + \frac{7}{19} a^{6} - \frac{5}{19} a^{5} + \frac{1}{19} a^{4} - \frac{6}{19} a^{3} + \frac{9}{19} a^{2} - \frac{6}{19} a + \frac{3}{19}$, $\frac{1}{52618858076972773} a^{24} - \frac{1184334960142648}{52618858076972773} a^{23} + \frac{438509416742639}{52618858076972773} a^{22} + \frac{1331309185323382}{52618858076972773} a^{21} - \frac{826988159815181}{52618858076972773} a^{20} - \frac{5813704846034538}{52618858076972773} a^{19} - \frac{17351980870480595}{52618858076972773} a^{18} + \frac{6534553150104855}{52618858076972773} a^{17} + \frac{259134087403138}{2287776438129251} a^{16} - \frac{608962458370800}{2287776438129251} a^{15} + \frac{18072306151323415}{52618858076972773} a^{14} + \frac{12488068659824691}{52618858076972773} a^{13} - \frac{10121233440274262}{52618858076972773} a^{12} + \frac{1141392841768167}{2287776438129251} a^{11} + \frac{20138301737042403}{52618858076972773} a^{10} - \frac{22104838068966824}{52618858076972773} a^{9} + \frac{4332144205198797}{52618858076972773} a^{8} + \frac{37962263285211}{2287776438129251} a^{7} - \frac{43407010630088}{52618858076972773} a^{6} + \frac{16499678470553920}{52618858076972773} a^{5} + \frac{11477558088807712}{52618858076972773} a^{4} - \frac{21569201750475004}{52618858076972773} a^{3} - \frac{12007795988191396}{52618858076972773} a^{2} + \frac{23317515040336254}{52618858076972773} a + \frac{10840594945188783}{52618858076972773}$

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 $2$)
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:  \( 87435.46969808154 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{5}\cdot(2\pi)^{10}\cdot 87435.46969808154 \cdot 1}{2\sqrt{722359806654877741268938522624}}\approx 0.157844526650959$ (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

\(\Q(\zeta_{11})^+\), 5.1.937024.1

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 R ${\href{/LocalNumberField/3.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{5}$

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
$2$2.5.0.1$x^{5} + x^{2} + 1$$1$$5$$0$$C_5$$[\ ]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.9$x^{10} - 6 x^{8} - 24 x^{6} + 80 x^{4} + 336 x^{2} + 33056$$2$$5$$15$$C_{10}$$[3]^{5}$
11Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.8.2t1.b.a$1$ $ 2^{3}$ \(\Q(\sqrt{-2}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.11.5t1.a.b$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
* 1.11.5t1.a.a$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.88.10t1.a.d$1$ $ 2^{3} \cdot 11 $ 10.0.7024111812608.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.c$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.88.10t1.a.c$1$ $ 2^{3} \cdot 11 $ 10.0.7024111812608.1 $C_{10}$ (as 10T1) $0$ $-1$
1.88.10t1.a.b$1$ $ 2^{3} \cdot 11 $ 10.0.7024111812608.1 $C_{10}$ (as 10T1) $0$ $-1$
* 1.11.5t1.a.d$1$ $ 11 $ \(\Q(\zeta_{11})^+\) $C_5$ (as 5T1) $0$ $1$
1.88.10t1.a.a$1$ $ 2^{3} \cdot 11 $ 10.0.7024111812608.1 $C_{10}$ (as 10T1) $0$ $-1$
* 2.968.10t6.a.c$2$ $ 2^{3} \cdot 11^{2}$ 25.5.722359806654877741268938522624.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.88.10t6.a.d$2$ $ 2^{3} \cdot 11 $ 25.5.722359806654877741268938522624.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.88.10t6.a.c$2$ $ 2^{3} \cdot 11 $ 25.5.722359806654877741268938522624.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.968.5t2.a.b$2$ $ 2^{3} \cdot 11^{2}$ 5.1.937024.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.968.5t2.a.a$2$ $ 2^{3} \cdot 11^{2}$ 5.1.937024.1 $D_{5}$ (as 5T2) $1$ $0$
* 2.88.10t6.a.a$2$ $ 2^{3} \cdot 11 $ 25.5.722359806654877741268938522624.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.968.10t6.a.d$2$ $ 2^{3} \cdot 11^{2}$ 25.5.722359806654877741268938522624.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.968.10t6.a.a$2$ $ 2^{3} \cdot 11^{2}$ 25.5.722359806654877741268938522624.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.968.10t6.a.b$2$ $ 2^{3} \cdot 11^{2}$ 25.5.722359806654877741268938522624.1 $C_5\times D_5$ (as 25T3) $0$ $0$
* 2.88.10t6.a.b$2$ $ 2^{3} \cdot 11 $ 25.5.722359806654877741268938522624.1 $C_5\times D_5$ (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 *.