Properties

Label 14.0.197...000.1
Degree $14$
Signature $[0, 7]$
Discriminant $-1.977\times 10^{21}$
Root discriminant \(33.20\)
Ramified primes $2,5,7$
Class number $1$
Class group trivial
Galois group $F_7$ (as 14T4)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 15*x^12 + 17*x^11 + 59*x^10 - 23*x^9 - 107*x^8 - 435*x^7 + 594*x^6 + 388*x^5 + 1158*x^4 - 408*x^3 + 250*x^2 + 6*x + 2)
 
gp: K = bnfinit(y^14 - y^13 - 15*y^12 + 17*y^11 + 59*y^10 - 23*y^9 - 107*y^8 - 435*y^7 + 594*y^6 + 388*y^5 + 1158*y^4 - 408*y^3 + 250*y^2 + 6*y + 2, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - x^13 - 15*x^12 + 17*x^11 + 59*x^10 - 23*x^9 - 107*x^8 - 435*x^7 + 594*x^6 + 388*x^5 + 1158*x^4 - 408*x^3 + 250*x^2 + 6*x + 2);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 - 15*x^12 + 17*x^11 + 59*x^10 - 23*x^9 - 107*x^8 - 435*x^7 + 594*x^6 + 388*x^5 + 1158*x^4 - 408*x^3 + 250*x^2 + 6*x + 2)
 

\( x^{14} - x^{13} - 15 x^{12} + 17 x^{11} + 59 x^{10} - 23 x^{9} - 107 x^{8} - 435 x^{7} + 594 x^{6} + 388 x^{5} + 1158 x^{4} - 408 x^{3} + 250 x^{2} + 6 x + 2 \) Copy content Toggle raw display

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

Invariants

Degree:  $14$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 7]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1977326743000000000000\) \(\medspace = -\,2^{12}\cdot 5^{12}\cdot 7^{11}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(33.20\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{6/7}5^{6/7}7^{5/6}\approx 36.42430080048362$
Ramified primes:   \(2\), \(5\), \(7\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-7}) \)
$\card{ \Aut(K/\Q) }$:  $2$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11}a^{11}+\frac{2}{11}a^{10}-\frac{4}{11}a^{9}-\frac{1}{11}a^{8}+\frac{4}{11}a^{7}+\frac{4}{11}a^{6}-\frac{4}{11}a^{5}+\frac{4}{11}a^{4}+\frac{5}{11}a^{3}+\frac{3}{11}a^{2}-\frac{5}{11}a-\frac{4}{11}$, $\frac{1}{473}a^{12}-\frac{21}{473}a^{11}+\frac{225}{473}a^{10}+\frac{157}{473}a^{9}+\frac{236}{473}a^{8}-\frac{2}{43}a^{7}+\frac{36}{473}a^{6}+\frac{96}{473}a^{5}-\frac{109}{473}a^{4}-\frac{178}{473}a^{3}-\frac{3}{11}a^{2}+\frac{199}{473}a-\frac{161}{473}$, $\frac{1}{13547955806627}a^{13}+\frac{12379551315}{13547955806627}a^{12}-\frac{137438584253}{13547955806627}a^{11}-\frac{4055119528776}{13547955806627}a^{10}+\frac{4455941572171}{13547955806627}a^{9}-\frac{3294739031500}{13547955806627}a^{8}-\frac{3017158066746}{13547955806627}a^{7}+\frac{412587467959}{1231632346057}a^{6}-\frac{4094210723176}{13547955806627}a^{5}+\frac{5407855341135}{13547955806627}a^{4}-\frac{4000871983637}{13547955806627}a^{3}+\frac{2163961037325}{13547955806627}a^{2}-\frac{258628756101}{13547955806627}a+\frac{584840440497}{13547955806627}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $6$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{249214540611}{13547955806627}a^{13}-\frac{156402356573}{13547955806627}a^{12}-\frac{3855707199669}{13547955806627}a^{11}+\frac{2900799658888}{13547955806627}a^{10}+\frac{16769529888435}{13547955806627}a^{9}-\frac{1168150292015}{13547955806627}a^{8}-\frac{31475389875043}{13547955806627}a^{7}-\frac{114195498259558}{13547955806627}a^{6}+\frac{113556649365682}{13547955806627}a^{5}+\frac{155258631781846}{13547955806627}a^{4}+\frac{302284720470060}{13547955806627}a^{3}-\frac{30789252160441}{13547955806627}a^{2}+\frac{41946477403143}{13547955806627}a+\frac{1170416907525}{13547955806627}$, $\frac{171171261033}{13547955806627}a^{13}+\frac{9786836236}{13547955806627}a^{12}-\frac{2649087916537}{13547955806627}a^{11}+\frac{76326058042}{13547955806627}a^{10}+\frac{11605557658300}{13547955806627}a^{9}+\frac{8900974043059}{13547955806627}a^{8}-\frac{15525888325062}{13547955806627}a^{7}-\frac{100203465601103}{13547955806627}a^{6}+\frac{8079453646958}{13547955806627}a^{5}+\frac{12846624071306}{1231632346057}a^{4}+\frac{340763708794609}{13547955806627}a^{3}+\frac{193123852199292}{13547955806627}a^{2}+\frac{1051966878287}{1231632346057}a+\frac{694327100451}{13547955806627}$, $\frac{3980419701685}{13547955806627}a^{13}-\frac{348424679334}{1231632346057}a^{12}-\frac{59968476603489}{13547955806627}a^{11}+\frac{65682364962404}{13547955806627}a^{10}+\frac{238977054232750}{13547955806627}a^{9}-\frac{86685464716791}{13547955806627}a^{8}-\frac{433966640482207}{13547955806627}a^{7}-\frac{17\!\cdots\!20}{13547955806627}a^{6}+\frac{23\!\cdots\!43}{13547955806627}a^{5}+\frac{16\!\cdots\!63}{13547955806627}a^{4}+\frac{45\!\cdots\!08}{13547955806627}a^{3}-\frac{14\!\cdots\!83}{13547955806627}a^{2}+\frac{874655056726277}{13547955806627}a+\frac{77682784133305}{13547955806627}$, $\frac{19462070487}{13547955806627}a^{13}+\frac{99421249347}{13547955806627}a^{12}+\frac{119540249788}{13547955806627}a^{11}-\frac{2840530433364}{13547955806627}a^{10}-\frac{3716825637179}{13547955806627}a^{9}+\frac{29438971644617}{13547955806627}a^{8}+\frac{505791729035}{1231632346057}a^{7}-\frac{96032986041455}{13547955806627}a^{6}-\frac{12420017447137}{13547955806627}a^{5}-\frac{42463057575091}{13547955806627}a^{4}+\frac{560364290938929}{13547955806627}a^{3}-\frac{190913550409546}{13547955806627}a^{2}+\frac{81338060676839}{13547955806627}a+\frac{10836939710757}{13547955806627}$, $\frac{10653201712321}{13547955806627}a^{13}-\frac{11322117245140}{13547955806627}a^{12}-\frac{3701685488233}{315068739689}a^{11}+\frac{4446076227163}{315068739689}a^{10}+\frac{617851254831488}{13547955806627}a^{9}-\frac{285311939817199}{13547955806627}a^{8}-\frac{11\!\cdots\!70}{13547955806627}a^{7}-\frac{45\!\cdots\!57}{13547955806627}a^{6}+\frac{66\!\cdots\!42}{13547955806627}a^{5}+\frac{37\!\cdots\!10}{13547955806627}a^{4}+\frac{12\!\cdots\!38}{13547955806627}a^{3}-\frac{51\!\cdots\!22}{13547955806627}a^{2}+\frac{29\!\cdots\!43}{13547955806627}a-\frac{88774256381765}{13547955806627}$, $\frac{3560896677709}{13547955806627}a^{13}-\frac{2040197050017}{13547955806627}a^{12}-\frac{55433524138309}{13547955806627}a^{11}+\frac{37550160608573}{13547955806627}a^{10}+\frac{243355571059108}{13547955806627}a^{9}+\frac{11008281649182}{13547955806627}a^{8}-\frac{446332032931366}{13547955806627}a^{7}-\frac{17\!\cdots\!14}{13547955806627}a^{6}+\frac{14\!\cdots\!66}{13547955806627}a^{5}+\frac{26\!\cdots\!19}{13547955806627}a^{4}+\frac{47\!\cdots\!84}{13547955806627}a^{3}-\frac{101951615464299}{13547955806627}a^{2}-\frac{12\!\cdots\!70}{13547955806627}a-\frac{56733543573087}{13547955806627}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 421014.85114896303 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 421014.85114896303 \cdot 1}{2\cdot\sqrt{1977326743000000000000}}\cr\approx \mathstrut & 1.83015249588756 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^14 - x^13 - 15*x^12 + 17*x^11 + 59*x^10 - 23*x^9 - 107*x^8 - 435*x^7 + 594*x^6 + 388*x^5 + 1158*x^4 - 408*x^3 + 250*x^2 + 6*x + 2)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^14 - x^13 - 15*x^12 + 17*x^11 + 59*x^10 - 23*x^9 - 107*x^8 - 435*x^7 + 594*x^6 + 388*x^5 + 1158*x^4 - 408*x^3 + 250*x^2 + 6*x + 2, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^14 - x^13 - 15*x^12 + 17*x^11 + 59*x^10 - 23*x^9 - 107*x^8 - 435*x^7 + 594*x^6 + 388*x^5 + 1158*x^4 - 408*x^3 + 250*x^2 + 6*x + 2);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - x^13 - 15*x^12 + 17*x^11 + 59*x^10 - 23*x^9 - 107*x^8 - 435*x^7 + 594*x^6 + 388*x^5 + 1158*x^4 - 408*x^3 + 250*x^2 + 6*x + 2);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$F_7$ (as 14T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

\(\Q(\sqrt{-7}) \), 7.1.16807000000.2

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: deg 42
Degree 7 sibling: 7.1.16807000000.2
Degree 21 sibling: deg 21
Minimal sibling: 7.1.16807000000.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ R R ${\href{/padicField/11.3.0.1}{3} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{7}$ ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.3.0.1}{3} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.7.0.1}{7} }^{2}$ ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.3.0.1}{3} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.2.0.1}{2} }^{7}$ ${\href{/padicField/43.1.0.1}{1} }^{14}$ ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$

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

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.7.6.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} + 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
\(5\) Copy content Toggle raw display 5.14.12.1$x^{14} + 28 x^{13} + 350 x^{12} + 2576 x^{11} + 12404 x^{10} + 41104 x^{9} + 96152 x^{8} + 160650 x^{7} + 192444 x^{6} + 165676 x^{5} + 106232 x^{4} + 65016 x^{3} + 59920 x^{2} + 57232 x + 27193$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
\(7\) Copy content Toggle raw display 7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 7$$6$$1$$5$$C_6$$[\ ]_{6}$