Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 - 7*x^12 + 133*x^11 - 49*x^10 - 1141*x^9 + 707*x^8 + 5731*x^7 - 2702*x^6 - 21658*x^5 + 11326*x^4 + 3178*x^3 + 45346*x^2 - 66094*x - 3442)
gp: K = bnfinit(y^14 - 7*y^13 - 7*y^12 + 133*y^11 - 49*y^10 - 1141*y^9 + 707*y^8 + 5731*y^7 - 2702*y^6 - 21658*y^5 + 11326*y^4 + 3178*y^3 + 45346*y^2 - 66094*y - 3442, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^14 - 7*x^13 - 7*x^12 + 133*x^11 - 49*x^10 - 1141*x^9 + 707*x^8 + 5731*x^7 - 2702*x^6 - 21658*x^5 + 11326*x^4 + 3178*x^3 + 45346*x^2 - 66094*x - 3442);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^14 - 7*x^13 - 7*x^12 + 133*x^11 - 49*x^10 - 1141*x^9 + 707*x^8 + 5731*x^7 - 2702*x^6 - 21658*x^5 + 11326*x^4 + 3178*x^3 + 45346*x^2 - 66094*x - 3442)
\( x^{14} - 7 x^{13} - 7 x^{12} + 133 x^{11} - 49 x^{10} - 1141 x^{9} + 707 x^{8} + 5731 x^{7} + \cdots - 3442 \)
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: | $[2, 6]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(252311969945749350012751872\)
\(\medspace = 2^{12}\cdot 3^{12}\cdot 7^{10}\cdot 17^{7}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(76.89\) | 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: | $2^{6/7}3^{6/7}7^{5/6}17^{1/2}\approx 96.93031261622481$ | ||
| Ramified primes: |
\(2\), \(3\), \(7\), \(17\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{10}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{3}a^{11}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{21}a^{12}+\frac{2}{21}a^{11}+\frac{2}{21}a^{10}+\frac{1}{7}a^{8}+\frac{2}{7}a^{7}-\frac{8}{21}a^{6}+\frac{5}{21}a^{5}+\frac{5}{21}a^{4}-\frac{1}{3}a^{3}+\frac{4}{21}a^{2}+\frac{1}{21}a-\frac{2}{7}$, $\frac{1}{14\!\cdots\!01}a^{13}-\frac{12\!\cdots\!66}{14\!\cdots\!01}a^{12}+\frac{39\!\cdots\!13}{14\!\cdots\!01}a^{11}-\frac{34\!\cdots\!89}{20\!\cdots\!43}a^{10}+\frac{12\!\cdots\!00}{48\!\cdots\!67}a^{9}-\frac{48\!\cdots\!37}{48\!\cdots\!67}a^{8}-\frac{22\!\cdots\!08}{48\!\cdots\!67}a^{7}+\frac{58\!\cdots\!98}{48\!\cdots\!67}a^{6}-\frac{14\!\cdots\!18}{14\!\cdots\!01}a^{5}-\frac{74\!\cdots\!17}{20\!\cdots\!43}a^{4}+\frac{11\!\cdots\!89}{48\!\cdots\!67}a^{3}-\frac{65\!\cdots\!04}{14\!\cdots\!01}a^{2}-\frac{50\!\cdots\!36}{14\!\cdots\!01}a-\frac{89\!\cdots\!22}{20\!\cdots\!43}$
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 $1$ (assuming GRH)
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: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| 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: |
$\frac{57\!\cdots\!58}{20\!\cdots\!43}a^{13}-\frac{28\!\cdots\!15}{14\!\cdots\!01}a^{12}-\frac{24\!\cdots\!44}{14\!\cdots\!01}a^{11}+\frac{17\!\cdots\!37}{48\!\cdots\!67}a^{10}-\frac{31\!\cdots\!48}{20\!\cdots\!43}a^{9}-\frac{13\!\cdots\!76}{48\!\cdots\!67}a^{8}+\frac{95\!\cdots\!50}{48\!\cdots\!67}a^{7}+\frac{18\!\cdots\!70}{14\!\cdots\!01}a^{6}-\frac{15\!\cdots\!91}{14\!\cdots\!01}a^{5}-\frac{22\!\cdots\!95}{48\!\cdots\!67}a^{4}+\frac{52\!\cdots\!93}{69\!\cdots\!81}a^{3}-\frac{59\!\cdots\!31}{14\!\cdots\!01}a^{2}+\frac{56\!\cdots\!04}{14\!\cdots\!01}a-\frac{81\!\cdots\!55}{14\!\cdots\!01}$, $\frac{73\!\cdots\!20}{69\!\cdots\!81}a^{13}-\frac{12\!\cdots\!96}{20\!\cdots\!43}a^{12}-\frac{11\!\cdots\!46}{69\!\cdots\!81}a^{11}+\frac{25\!\cdots\!45}{20\!\cdots\!43}a^{10}+\frac{89\!\cdots\!93}{69\!\cdots\!81}a^{9}-\frac{77\!\cdots\!61}{69\!\cdots\!81}a^{8}-\frac{62\!\cdots\!42}{69\!\cdots\!81}a^{7}+\frac{38\!\cdots\!84}{69\!\cdots\!81}a^{6}+\frac{37\!\cdots\!40}{69\!\cdots\!81}a^{5}-\frac{36\!\cdots\!39}{20\!\cdots\!43}a^{4}-\frac{11\!\cdots\!43}{69\!\cdots\!81}a^{3}-\frac{11\!\cdots\!61}{69\!\cdots\!81}a^{2}+\frac{77\!\cdots\!34}{20\!\cdots\!43}a-\frac{49\!\cdots\!31}{20\!\cdots\!43}$, $\frac{52\!\cdots\!68}{69\!\cdots\!81}a^{13}-\frac{34\!\cdots\!48}{69\!\cdots\!81}a^{12}-\frac{57\!\cdots\!56}{69\!\cdots\!81}a^{11}+\frac{68\!\cdots\!80}{69\!\cdots\!81}a^{10}+\frac{14\!\cdots\!24}{69\!\cdots\!81}a^{9}-\frac{62\!\cdots\!86}{69\!\cdots\!81}a^{8}+\frac{17\!\cdots\!24}{69\!\cdots\!81}a^{7}+\frac{33\!\cdots\!88}{69\!\cdots\!81}a^{6}+\frac{50\!\cdots\!00}{69\!\cdots\!81}a^{5}-\frac{12\!\cdots\!24}{69\!\cdots\!81}a^{4}-\frac{22\!\cdots\!00}{69\!\cdots\!81}a^{3}+\frac{70\!\cdots\!04}{69\!\cdots\!81}a^{2}+\frac{40\!\cdots\!16}{69\!\cdots\!81}a+\frac{11\!\cdots\!95}{69\!\cdots\!81}$, $\frac{22\!\cdots\!38}{69\!\cdots\!81}a^{13}-\frac{88\!\cdots\!28}{48\!\cdots\!67}a^{12}-\frac{20\!\cdots\!72}{48\!\cdots\!67}a^{11}+\frac{50\!\cdots\!13}{14\!\cdots\!01}a^{10}+\frac{49\!\cdots\!99}{20\!\cdots\!43}a^{9}-\frac{13\!\cdots\!94}{48\!\cdots\!67}a^{8}-\frac{56\!\cdots\!54}{48\!\cdots\!67}a^{7}+\frac{17\!\cdots\!41}{14\!\cdots\!01}a^{6}+\frac{27\!\cdots\!59}{48\!\cdots\!67}a^{5}-\frac{49\!\cdots\!86}{14\!\cdots\!01}a^{4}+\frac{10\!\cdots\!56}{20\!\cdots\!43}a^{3}-\frac{47\!\cdots\!23}{48\!\cdots\!67}a^{2}-\frac{28\!\cdots\!27}{14\!\cdots\!01}a-\frac{53\!\cdots\!43}{48\!\cdots\!67}$, $\frac{10\!\cdots\!60}{14\!\cdots\!01}a^{13}-\frac{71\!\cdots\!48}{14\!\cdots\!01}a^{12}-\frac{58\!\cdots\!82}{14\!\cdots\!01}a^{11}+\frac{19\!\cdots\!51}{20\!\cdots\!43}a^{10}-\frac{24\!\cdots\!77}{48\!\cdots\!67}a^{9}-\frac{37\!\cdots\!35}{48\!\cdots\!67}a^{8}+\frac{92\!\cdots\!18}{14\!\cdots\!01}a^{7}+\frac{18\!\cdots\!78}{48\!\cdots\!67}a^{6}-\frac{38\!\cdots\!85}{14\!\cdots\!01}a^{5}-\frac{10\!\cdots\!53}{69\!\cdots\!81}a^{4}+\frac{52\!\cdots\!69}{48\!\cdots\!67}a^{3}+\frac{28\!\cdots\!00}{14\!\cdots\!01}a^{2}+\frac{43\!\cdots\!90}{14\!\cdots\!01}a-\frac{11\!\cdots\!69}{20\!\cdots\!43}$, $\frac{11\!\cdots\!00}{14\!\cdots\!01}a^{13}-\frac{36\!\cdots\!24}{14\!\cdots\!01}a^{12}-\frac{80\!\cdots\!65}{48\!\cdots\!67}a^{11}+\frac{61\!\cdots\!73}{14\!\cdots\!01}a^{10}+\frac{21\!\cdots\!94}{14\!\cdots\!01}a^{9}-\frac{46\!\cdots\!27}{14\!\cdots\!01}a^{8}-\frac{39\!\cdots\!83}{48\!\cdots\!67}a^{7}+\frac{17\!\cdots\!56}{14\!\cdots\!01}a^{6}+\frac{48\!\cdots\!94}{14\!\cdots\!01}a^{5}-\frac{39\!\cdots\!47}{14\!\cdots\!01}a^{4}-\frac{21\!\cdots\!83}{14\!\cdots\!01}a^{3}-\frac{84\!\cdots\!31}{14\!\cdots\!01}a^{2}+\frac{18\!\cdots\!93}{14\!\cdots\!01}a-\frac{11\!\cdots\!93}{14\!\cdots\!01}$, $\frac{20\!\cdots\!35}{14\!\cdots\!01}a^{13}-\frac{13\!\cdots\!86}{48\!\cdots\!67}a^{12}-\frac{69\!\cdots\!51}{14\!\cdots\!01}a^{11}+\frac{10\!\cdots\!93}{14\!\cdots\!01}a^{10}+\frac{34\!\cdots\!54}{48\!\cdots\!67}a^{9}-\frac{97\!\cdots\!47}{14\!\cdots\!01}a^{8}-\frac{84\!\cdots\!98}{14\!\cdots\!01}a^{7}+\frac{31\!\cdots\!34}{14\!\cdots\!01}a^{6}+\frac{12\!\cdots\!57}{48\!\cdots\!67}a^{5}+\frac{27\!\cdots\!33}{48\!\cdots\!67}a^{4}-\frac{86\!\cdots\!32}{14\!\cdots\!01}a^{3}-\frac{18\!\cdots\!71}{14\!\cdots\!01}a^{2}-\frac{12\!\cdots\!73}{14\!\cdots\!01}a-\frac{58\!\cdots\!65}{14\!\cdots\!01}$
| 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: | \( 127212203.73823108 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 127212203.73823108 \cdot 1}{2\cdot\sqrt{252311969945749350012751872}}\cr\approx \mathstrut & 0.985528197697170 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^14 - 7*x^13 - 7*x^12 + 133*x^11 - 49*x^10 - 1141*x^9 + 707*x^8 + 5731*x^7 - 2702*x^6 - 21658*x^5 + 11326*x^4 + 3178*x^3 + 45346*x^2 - 66094*x - 3442)
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 - 7*x^13 - 7*x^12 + 133*x^11 - 49*x^10 - 1141*x^9 + 707*x^8 + 5731*x^7 - 2702*x^6 - 21658*x^5 + 11326*x^4 + 3178*x^3 + 45346*x^2 - 66094*x - 3442, 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 - 7*x^13 - 7*x^12 + 133*x^11 - 49*x^10 - 1141*x^9 + 707*x^8 + 5731*x^7 - 2702*x^6 - 21658*x^5 + 11326*x^4 + 3178*x^3 + 45346*x^2 - 66094*x - 3442);
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 - 7*x^13 - 7*x^12 + 133*x^11 - 49*x^10 - 1141*x^9 + 707*x^8 + 5731*x^7 - 2702*x^6 - 21658*x^5 + 11326*x^4 + 3178*x^3 + 45346*x^2 - 66094*x - 3442);
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
$C_2\times F_7$ (as 14T7):
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 84 |
| The 14 conjugacy class representatives for $F_7 \times C_2$ |
| Character table for $F_7 \times C_2$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), 7.1.784147392.1 |
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
| Degree 14 sibling: | data not computed |
| Degree 28 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.6.0.1}{6} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{6}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.6.0.1}{6} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.14.0.1}{14} }$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{7}$ | ${\href{/padicField/43.7.0.1}{7} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ |
| 2.7.6.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $[\ ]_{7}^{3}$ | |
|
\(3\)
| 3.14.12.1 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1484 x^{10} + 3752 x^{9} + 7448 x^{8} + 11782 x^{7} + 14938 x^{6} + 15008 x^{5} + 11452 x^{4} + 6328 x^{3} + 2632 x^{2} + 896 x + 185$ | $7$ | $2$ | $12$ | $F_7$ | $[\ ]_{7}^{6}$ |
|
\(7\)
| 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
|
\(17\)
| 17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 17.12.6.1 | $x^{12} + 918 x^{11} + 351241 x^{10} + 71712630 x^{9} + 8244584136 x^{8} + 506874732756 x^{7} + 13125344775560 x^{6} + 9625198256031 x^{5} + 28457943732288 x^{4} + 16844354225613 x^{3} + 132306217741765 x^{2} + 68598705820311 x + 44162739951115$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |