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Properties

Label 42.0.475...976.1
Degree $42$
Signature $[0, 21]$
Discriminant $-4.751\times 10^{69}$
Root discriminant \(45.60\)
Ramified primes $2,29$
Class number not computed
Class group not computed
Galois group $S_3\times C_7$ (as 42T6)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29)
 
gp: K = bnfinit(y^42 + 46*y^40 + 825*y^38 + 9020*y^36 + 65490*y^34 + 325184*y^32 + 1115476*y^30 + 2295041*y^28 + 2315947*y^26 - 2734559*y^24 - 14848773*y^22 - 6221636*y^20 + 35462359*y^18 - 4544065*y^16 - 16428325*y^14 + 5957296*y^12 + 1133059*y^10 - 497379*y^8 - 50982*y^6 + 12238*y^4 + 1537*y^2 + 29, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29)
 

\( x^{42} + 46 x^{40} + 825 x^{38} + 9020 x^{36} + 65490 x^{34} + 325184 x^{32} + 1115476 x^{30} + \cdots + 29 \) Copy content Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 21]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-4750963959983097206493997106048442150328763366840752912458449709694976\) \(\medspace = -\,2^{42}\cdot 29^{39}\) 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:  \(45.60\)
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:  $2\cdot 29^{13/14}\approx 45.60064585163551$
Ramified primes:   \(2\), \(29\) 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:  \(\Q(\sqrt{-29}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $\frac{1}{4}a^{28}-\frac{1}{2}a^{27}+\frac{1}{4}a^{26}-\frac{1}{2}a^{25}-\frac{1}{2}a^{24}+\frac{1}{4}a^{22}-\frac{1}{2}a^{19}-\frac{1}{4}a^{18}-\frac{1}{2}a^{17}+\frac{1}{4}a^{16}-\frac{1}{4}a^{14}+\frac{1}{4}a^{12}-\frac{1}{2}a^{11}+\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{29}+\frac{1}{4}a^{27}-\frac{1}{2}a^{25}+\frac{1}{4}a^{23}-\frac{1}{2}a^{22}-\frac{1}{2}a^{20}-\frac{1}{4}a^{19}+\frac{1}{4}a^{17}-\frac{1}{2}a^{16}-\frac{1}{4}a^{15}-\frac{1}{2}a^{14}+\frac{1}{4}a^{13}+\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{30}-\frac{1}{2}a^{27}+\frac{1}{4}a^{26}-\frac{1}{2}a^{25}-\frac{1}{4}a^{24}-\frac{1}{2}a^{23}-\frac{1}{4}a^{22}-\frac{1}{2}a^{21}-\frac{1}{4}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{31}+\frac{1}{4}a^{27}-\frac{1}{4}a^{25}-\frac{1}{2}a^{24}-\frac{1}{4}a^{23}-\frac{1}{4}a^{21}-\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}-\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}-\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{32}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{1}{4}a^{24}-\frac{1}{2}a^{22}-\frac{1}{2}a^{21}-\frac{1}{2}a^{20}-\frac{1}{4}a^{18}-\frac{1}{2}a^{17}+\frac{1}{4}a^{16}-\frac{1}{2}a^{15}+\frac{1}{4}a^{14}-\frac{1}{2}a^{13}+\frac{1}{4}a^{12}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}$, $\frac{1}{4}a^{33}-\frac{1}{2}a^{27}-\frac{1}{2}a^{26}+\frac{1}{4}a^{25}-\frac{1}{2}a^{23}-\frac{1}{2}a^{21}-\frac{1}{4}a^{19}+\frac{1}{4}a^{17}+\frac{1}{4}a^{15}+\frac{1}{4}a^{13}-\frac{1}{2}a^{12}+\frac{1}{4}a^{11}+\frac{1}{4}a^{9}+\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{34}-\frac{1}{2}a^{27}-\frac{1}{4}a^{26}-\frac{1}{2}a^{24}-\frac{1}{4}a^{20}-\frac{1}{4}a^{18}-\frac{1}{4}a^{16}-\frac{1}{4}a^{14}-\frac{1}{2}a^{13}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{35}-\frac{1}{4}a^{27}-\frac{1}{2}a^{26}-\frac{1}{2}a^{25}-\frac{1}{2}a^{22}-\frac{1}{4}a^{21}-\frac{1}{4}a^{19}-\frac{1}{2}a^{18}-\frac{1}{4}a^{17}-\frac{1}{2}a^{16}-\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{2}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{764}a^{36}-\frac{9}{764}a^{34}+\frac{7}{191}a^{32}+\frac{83}{764}a^{30}+\frac{2}{191}a^{28}-\frac{233}{764}a^{26}+\frac{145}{764}a^{24}-\frac{303}{764}a^{22}-\frac{1}{2}a^{21}+\frac{57}{764}a^{20}-\frac{1}{2}a^{19}+\frac{205}{764}a^{18}-\frac{353}{764}a^{16}-\frac{1}{2}a^{15}+\frac{61}{764}a^{14}-\frac{83}{764}a^{12}-\frac{49}{764}a^{10}-\frac{1}{2}a^{9}-\frac{127}{764}a^{8}+\frac{70}{191}a^{6}-\frac{1}{2}a^{5}+\frac{139}{382}a^{4}-\frac{1}{2}a^{3}-\frac{3}{382}a^{2}-\frac{1}{2}a+\frac{12}{191}$, $\frac{1}{764}a^{37}-\frac{9}{764}a^{35}+\frac{7}{191}a^{33}+\frac{83}{764}a^{31}+\frac{2}{191}a^{29}-\frac{233}{764}a^{27}+\frac{145}{764}a^{25}-\frac{303}{764}a^{23}-\frac{1}{2}a^{22}+\frac{57}{764}a^{21}-\frac{1}{2}a^{20}+\frac{205}{764}a^{19}-\frac{353}{764}a^{17}-\frac{1}{2}a^{16}+\frac{61}{764}a^{15}-\frac{83}{764}a^{13}-\frac{49}{764}a^{11}-\frac{1}{2}a^{10}-\frac{127}{764}a^{9}+\frac{70}{191}a^{7}-\frac{1}{2}a^{6}+\frac{139}{382}a^{5}-\frac{1}{2}a^{4}-\frac{3}{382}a^{3}-\frac{1}{2}a^{2}+\frac{12}{191}a$, $\frac{1}{764}a^{38}-\frac{53}{764}a^{34}-\frac{47}{764}a^{32}-\frac{9}{764}a^{30}+\frac{15}{382}a^{28}-\frac{1}{2}a^{27}-\frac{233}{764}a^{26}-\frac{1}{2}a^{25}+\frac{119}{382}a^{24}-\frac{1}{2}a^{23}-\frac{187}{764}a^{22}-\frac{23}{382}a^{20}+\frac{155}{764}a^{18}-\frac{251}{764}a^{16}-\frac{1}{2}a^{15}-\frac{107}{764}a^{14}-\frac{223}{764}a^{12}+\frac{5}{764}a^{10}-\frac{1}{2}a^{9}+\frac{23}{191}a^{8}+\frac{315}{764}a^{6}+\frac{13}{764}a^{4}-\frac{197}{764}a^{2}+\frac{241}{764}$, $\frac{1}{764}a^{39}-\frac{53}{764}a^{35}-\frac{47}{764}a^{33}-\frac{9}{764}a^{31}+\frac{15}{382}a^{29}-\frac{233}{764}a^{27}+\frac{119}{382}a^{25}-\frac{1}{2}a^{24}-\frac{187}{764}a^{23}-\frac{1}{2}a^{22}-\frac{23}{382}a^{21}+\frac{155}{764}a^{19}-\frac{1}{2}a^{18}-\frac{251}{764}a^{17}-\frac{107}{764}a^{15}-\frac{1}{2}a^{14}-\frac{223}{764}a^{13}-\frac{1}{2}a^{12}+\frac{5}{764}a^{11}+\frac{23}{191}a^{9}-\frac{1}{2}a^{8}+\frac{315}{764}a^{7}-\frac{1}{2}a^{6}+\frac{13}{764}a^{5}-\frac{1}{2}a^{4}-\frac{197}{764}a^{3}-\frac{1}{2}a^{2}+\frac{241}{764}a-\frac{1}{2}$, $\frac{1}{25\!\cdots\!76}a^{40}+\frac{98\!\cdots\!35}{25\!\cdots\!76}a^{38}-\frac{66\!\cdots\!01}{12\!\cdots\!38}a^{36}-\frac{24\!\cdots\!76}{36\!\cdots\!57}a^{34}+\frac{30\!\cdots\!99}{62\!\cdots\!69}a^{32}+\frac{15\!\cdots\!79}{25\!\cdots\!76}a^{30}+\frac{26\!\cdots\!82}{36\!\cdots\!57}a^{28}+\frac{70\!\cdots\!05}{25\!\cdots\!76}a^{26}-\frac{87\!\cdots\!55}{25\!\cdots\!76}a^{24}-\frac{1}{2}a^{23}+\frac{12\!\cdots\!55}{62\!\cdots\!69}a^{22}+\frac{96\!\cdots\!44}{62\!\cdots\!69}a^{20}-\frac{49\!\cdots\!99}{25\!\cdots\!76}a^{18}-\frac{1}{2}a^{17}-\frac{10\!\cdots\!77}{25\!\cdots\!76}a^{16}+\frac{10\!\cdots\!11}{25\!\cdots\!76}a^{14}-\frac{11\!\cdots\!45}{25\!\cdots\!76}a^{12}-\frac{1}{2}a^{11}+\frac{11\!\cdots\!31}{12\!\cdots\!38}a^{10}-\frac{1}{2}a^{9}+\frac{42\!\cdots\!27}{12\!\cdots\!38}a^{8}+\frac{19\!\cdots\!03}{12\!\cdots\!38}a^{6}-\frac{1}{2}a^{5}-\frac{66\!\cdots\!23}{25\!\cdots\!76}a^{4}-\frac{1}{2}a^{3}-\frac{28\!\cdots\!62}{62\!\cdots\!69}a^{2}-\frac{43\!\cdots\!89}{25\!\cdots\!76}$, $\frac{1}{25\!\cdots\!76}a^{41}+\frac{98\!\cdots\!35}{25\!\cdots\!76}a^{39}-\frac{66\!\cdots\!01}{12\!\cdots\!38}a^{37}-\frac{24\!\cdots\!76}{36\!\cdots\!57}a^{35}+\frac{30\!\cdots\!99}{62\!\cdots\!69}a^{33}+\frac{15\!\cdots\!79}{25\!\cdots\!76}a^{31}+\frac{26\!\cdots\!82}{36\!\cdots\!57}a^{29}+\frac{70\!\cdots\!05}{25\!\cdots\!76}a^{27}-\frac{87\!\cdots\!55}{25\!\cdots\!76}a^{25}-\frac{1}{2}a^{24}+\frac{12\!\cdots\!55}{62\!\cdots\!69}a^{23}+\frac{96\!\cdots\!44}{62\!\cdots\!69}a^{21}-\frac{49\!\cdots\!99}{25\!\cdots\!76}a^{19}-\frac{1}{2}a^{18}-\frac{10\!\cdots\!77}{25\!\cdots\!76}a^{17}+\frac{10\!\cdots\!11}{25\!\cdots\!76}a^{15}-\frac{11\!\cdots\!45}{25\!\cdots\!76}a^{13}-\frac{1}{2}a^{12}+\frac{11\!\cdots\!31}{12\!\cdots\!38}a^{11}-\frac{1}{2}a^{10}+\frac{42\!\cdots\!27}{12\!\cdots\!38}a^{9}+\frac{19\!\cdots\!03}{12\!\cdots\!38}a^{7}-\frac{1}{2}a^{6}-\frac{66\!\cdots\!23}{25\!\cdots\!76}a^{5}-\frac{1}{2}a^{4}-\frac{28\!\cdots\!62}{62\!\cdots\!69}a^{3}-\frac{43\!\cdots\!89}{25\!\cdots\!76}a$ 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

not computed

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:  $20$
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:  not computed
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:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29)
 
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^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29, 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^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29);
 
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^42 + 46*x^40 + 825*x^38 + 9020*x^36 + 65490*x^34 + 325184*x^32 + 1115476*x^30 + 2295041*x^28 + 2315947*x^26 - 2734559*x^24 - 14848773*x^22 - 6221636*x^20 + 35462359*x^18 - 4544065*x^16 - 16428325*x^14 + 5957296*x^12 + 1133059*x^10 - 497379*x^8 - 50982*x^6 + 12238*x^4 + 1537*x^2 + 29);
 
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

$S_3\times C_7$ (as 42T6):

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 21 conjugacy class representatives for $S_3\times C_7$
Character table for $S_3\times C_7$ is not computed

Intermediate fields

\(\Q(\sqrt{-29}) \), 3.1.116.1 x3, 6.0.1560896.1, 7.7.594823321.1, 14.0.168110140833113738264576.1, 21.7.99995832264130420565259872976896.1 x3

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 21 sibling: data not computed
Minimal sibling: 21.7.99995832264130420565259872976896.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $21^{2}$ $21^{2}$ ${\href{/padicField/7.14.0.1}{14} }^{3}$ $21^{2}$ $21^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{21}$ ${\href{/padicField/19.7.0.1}{7} }^{6}$ ${\href{/padicField/23.14.0.1}{14} }^{3}$ R $21^{2}$ ${\href{/padicField/37.14.0.1}{14} }^{3}$ ${\href{/padicField/41.2.0.1}{2} }^{21}$ $21^{2}$ $21^{2}$ $21^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{21}$

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 2.14.14.15$x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.15$x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$$2$$7$$14$$C_{14}$$[2]^{7}$
2.14.14.15$x^{14} + 14 x^{13} + 126 x^{12} + 784 x^{11} + 4300 x^{10} + 19592 x^{9} + 80680 x^{8} + 276608 x^{7} + 822832 x^{6} + 1982880 x^{5} + 3998112 x^{4} + 6222080 x^{3} + 7679040 x^{2} + 6275456 x + 3453824$$2$$7$$14$$C_{14}$$[2]^{7}$
\(29\) Copy content Toggle raw display 29.14.13.1$x^{14} + 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} + 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$
29.14.13.1$x^{14} + 29$$14$$1$$13$$C_{14}$$[\ ]_{14}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.116.2t1.a.a$1$ $ 2^{2} \cdot 29 $ \(\Q(\sqrt{-29}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.29.7t1.a.a$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.a$1$ $ 2^{2} \cdot 29 $ 14.0.168110140833113738264576.1 $C_{14}$ (as 14T1) $0$ $-1$
* 1.116.14t1.a.b$1$ $ 2^{2} \cdot 29 $ 14.0.168110140833113738264576.1 $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.b$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
* 1.29.7t1.a.c$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
* 1.29.7t1.a.d$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.c$1$ $ 2^{2} \cdot 29 $ 14.0.168110140833113738264576.1 $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.e$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
* 1.116.14t1.a.d$1$ $ 2^{2} \cdot 29 $ 14.0.168110140833113738264576.1 $C_{14}$ (as 14T1) $0$ $-1$
* 1.116.14t1.a.e$1$ $ 2^{2} \cdot 29 $ 14.0.168110140833113738264576.1 $C_{14}$ (as 14T1) $0$ $-1$
* 1.116.14t1.a.f$1$ $ 2^{2} \cdot 29 $ 14.0.168110140833113738264576.1 $C_{14}$ (as 14T1) $0$ $-1$
* 1.29.7t1.a.f$1$ $ 29 $ 7.7.594823321.1 $C_7$ (as 7T1) $0$ $1$
*2 2.116.3t2.a.a$2$ $ 2^{2} \cdot 29 $ 3.1.116.1 $S_3$ (as 3T2) $1$ $0$
*2 2.3364.21t6.b.a$2$ $ 2^{2} \cdot 29^{2}$ 42.0.4750963959983097206493997106048442150328763366840752912458449709694976.1 $S_3\times C_7$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.b$2$ $ 2^{2} \cdot 29^{2}$ 42.0.4750963959983097206493997106048442150328763366840752912458449709694976.1 $S_3\times C_7$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.c$2$ $ 2^{2} \cdot 29^{2}$ 42.0.4750963959983097206493997106048442150328763366840752912458449709694976.1 $S_3\times C_7$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.d$2$ $ 2^{2} \cdot 29^{2}$ 42.0.4750963959983097206493997106048442150328763366840752912458449709694976.1 $S_3\times C_7$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.e$2$ $ 2^{2} \cdot 29^{2}$ 42.0.4750963959983097206493997106048442150328763366840752912458449709694976.1 $S_3\times C_7$ (as 42T6) $0$ $0$
*2 2.3364.21t6.b.f$2$ $ 2^{2} \cdot 29^{2}$ 42.0.4750963959983097206493997106048442150328763366840752912458449709694976.1 $S_3\times C_7$ (as 42T6) $0$ $0$

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 *.