↑

Properties

Label 15.15.2431345955...5625.1
Degree $15$
Signature $[15, 0]$
Discriminant $5^{18}\cdot 7^{10}\cdot 41^{12}$
Root discriminant $492.48$
Ramified primes $5, 7, 41$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $(C_5^2 : C_3):C_2$ (as 15T12)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9370162894441, 245238148250, 2260510932180, 170114602855, -172685011550, -21713100456, 5416669490, 868413005, -78738040, -15801400, 519839, 142680, -1230, -615, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^15 - 615*x^13 - 1230*x^12 + 142680*x^11 + 519839*x^10 - 15801400*x^9 - 78738040*x^8 + 868413005*x^7 + 5416669490*x^6 - 21713100456*x^5 - 172685011550*x^4 + 170114602855*x^3 + 2260510932180*x^2 + 245238148250*x - 9370162894441)
 
gp: K = bnfinit(x^15 - 615*x^13 - 1230*x^12 + 142680*x^11 + 519839*x^10 - 15801400*x^9 - 78738040*x^8 + 868413005*x^7 + 5416669490*x^6 - 21713100456*x^5 - 172685011550*x^4 + 170114602855*x^3 + 2260510932180*x^2 + 245238148250*x - 9370162894441, 1)
 

Normalized defining polynomial

\( x^{15} - 615 x^{13} - 1230 x^{12} + 142680 x^{11} + 519839 x^{10} - 15801400 x^{9} - 78738040 x^{8} + 868413005 x^{7} + 5416669490 x^{6} - 21713100456 x^{5} - 172685011550 x^{4} + 170114602855 x^{3} + 2260510932180 x^{2} + 245238148250 x - 9370162894441 \)

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:  \(24313459552402584855693089176177978515625=5^{18}\cdot 7^{10}\cdot 41^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $492.48$
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:  $5, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{41} a^{5}$, $\frac{1}{41} a^{6}$, $\frac{1}{41} a^{7}$, $\frac{1}{41} a^{8}$, $\frac{1}{41} a^{9}$, $\frac{1}{1681} a^{10}$, $\frac{1}{1681} a^{11}$, $\frac{1}{1681} a^{12}$, $\frac{1}{3362} a^{13} - \frac{1}{82} a^{8} - \frac{1}{82} a^{6} - \frac{1}{82} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{14587786527303725826177384554872086620835588668} a^{14} - \frac{7107596608649651793341984807790826302353}{339250849472179670376218245462141549321757876} a^{13} - \frac{1762993889527640161964647148422822130742143}{7293893263651862913088692277436043310417794334} a^{12} + \frac{281068939449035875259003493708389117587476}{3646946631825931456544346138718021655208897167} a^{11} - \frac{438480756754574048567896625540888484090344}{3646946631825931456544346138718021655208897167} a^{10} + \frac{1636403153241121923787933255159764548378979}{355799671397651849418960598899319185874038748} a^{9} - \frac{4092907353950302443294310465964592283282553}{355799671397651849418960598899319185874038748} a^{8} - \frac{2935479740683775798729421356940436838315753}{355799671397651849418960598899319185874038748} a^{7} - \frac{525448753204734231423767964702183704484695}{88949917849412962354740149724829796468509687} a^{6} + \frac{108275400545678687392938806800240500487115}{177899835698825924709480299449659592937019374} a^{5} + \frac{191806176146816828229116202503546425849363}{4339020382898193285597080474381941291146814} a^{4} - \frac{903855338610048771541299197897227593895160}{2169510191449096642798540237190970645573407} a^{3} + \frac{4309603145162236915546042541690796882450383}{8678040765796386571194160948763882582293628} a^{2} + \frac{2862066925944148258843069800142205799417775}{8678040765796386571194160948763882582293628} a + \frac{55154863202252132392149228310782287601091}{201814901530148524911492115087532153076596}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  \( 676079996900000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5^2:C_6$ (as 15T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 150
The 10 conjugacy class representatives for $(C_5^2 : C_3):C_2$
Character table for $(C_5^2 : C_3):C_2$

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 15 sibling: data not computed
Degree 25 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{5}$

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
5Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$41$41.5.4.3$x^{5} - 1476$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.2$x^{5} + 246$$5$$1$$4$$C_5$$[\ ]_{5}$
41.5.4.4$x^{5} + 8856$$5$$1$$4$$C_5$$[\ ]_{5}$