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Properties

Label 16.0.210...625.1
Degree $16$
Signature $[0, 8]$
Discriminant $2.107\times 10^{27}$
Root discriminant \(51.02\)
Ramified primes $5,149$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^3:C_4$ (as 16T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869)
 
gp: K = bnfinit(y^16 - 4*y^15 + 56*y^14 - 182*y^13 + 1092*y^12 - 3072*y^11 + 9994*y^10 - 20362*y^9 + 40781*y^8 - 34526*y^7 + 13090*y^6 + 38884*y^5 + 15301*y^4 - 14428*y^3 + 40421*y^2 + 9424*y + 27869, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869)
 

\( x^{16} - 4 x^{15} + 56 x^{14} - 182 x^{13} + 1092 x^{12} - 3072 x^{11} + 9994 x^{10} - 20362 x^{9} + \cdots + 27869 \) Copy content Toggle raw display

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(2106797133618518949609765625\) \(\medspace = 5^{8}\cdot 149^{10}\) 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:  \(51.02\)
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:  $5^{1/2}149^{3/4}\approx 95.3618578558619$
Ramified primes:   \(5\), \(149\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $8$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{10}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{7}+\frac{1}{10}a^{6}-\frac{3}{10}a^{4}-\frac{3}{10}a^{3}+\frac{1}{10}a^{2}+\frac{2}{5}$, $\frac{1}{10}a^{11}+\frac{1}{10}a^{9}+\frac{1}{10}a^{8}-\frac{1}{10}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{3}{10}$, $\frac{1}{580}a^{12}-\frac{23}{580}a^{11}-\frac{13}{580}a^{10}-\frac{13}{116}a^{9}+\frac{3}{290}a^{8}-\frac{233}{580}a^{7}-\frac{69}{145}a^{6}+\frac{177}{580}a^{5}+\frac{15}{58}a^{4}-\frac{121}{580}a^{3}-\frac{99}{580}a^{2}+\frac{14}{29}a-\frac{3}{20}$, $\frac{1}{580}a^{13}-\frac{1}{29}a^{11}-\frac{4}{145}a^{10}+\frac{19}{580}a^{9}+\frac{137}{580}a^{8}-\frac{9}{580}a^{7}+\frac{93}{580}a^{6}-\frac{49}{116}a^{5}-\frac{209}{580}a^{4}-\frac{68}{145}a^{3}+\frac{207}{580}a^{2}-\frac{259}{580}a-\frac{1}{4}$, $\frac{1}{259260}a^{14}+\frac{1}{12963}a^{13}+\frac{47}{129630}a^{12}+\frac{5557}{129630}a^{11}-\frac{1679}{51852}a^{10}+\frac{10523}{51852}a^{9}+\frac{48017}{259260}a^{8}+\frac{106867}{259260}a^{7}-\frac{5091}{17284}a^{6}+\frac{16391}{86420}a^{5}+\frac{8683}{25926}a^{4}-\frac{66587}{259260}a^{3}-\frac{19617}{86420}a^{2}+\frac{99733}{259260}a+\frac{913}{2235}$, $\frac{1}{71\!\cdots\!00}a^{15}-\frac{31\!\cdots\!77}{23\!\cdots\!00}a^{14}+\frac{32\!\cdots\!31}{23\!\cdots\!00}a^{13}-\frac{22\!\cdots\!74}{59\!\cdots\!25}a^{12}+\frac{33\!\cdots\!43}{71\!\cdots\!00}a^{11}-\frac{24\!\cdots\!39}{59\!\cdots\!25}a^{10}-\frac{18\!\cdots\!31}{47\!\cdots\!60}a^{9}+\frac{79\!\cdots\!79}{82\!\cdots\!00}a^{8}-\frac{45\!\cdots\!33}{14\!\cdots\!80}a^{7}+\frac{99\!\cdots\!93}{23\!\cdots\!00}a^{6}-\frac{54\!\cdots\!99}{35\!\cdots\!50}a^{5}-\frac{35\!\cdots\!79}{71\!\cdots\!90}a^{4}+\frac{28\!\cdots\!69}{17\!\cdots\!75}a^{3}+\frac{43\!\cdots\!87}{14\!\cdots\!80}a^{2}+\frac{27\!\cdots\!93}{59\!\cdots\!25}a+\frac{16\!\cdots\!93}{79\!\cdots\!00}$ Copy content Toggle raw display

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

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

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 $2$) 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:   $\frac{8204825633263}{48\!\cdots\!30}a^{15}-\frac{13938946713497}{19\!\cdots\!32}a^{14}+\frac{187376413877361}{19\!\cdots\!32}a^{13}-\frac{649887524859219}{19\!\cdots\!32}a^{12}+\frac{37\!\cdots\!73}{19\!\cdots\!32}a^{11}-\frac{28\!\cdots\!50}{48\!\cdots\!33}a^{10}+\frac{18\!\cdots\!07}{97\!\cdots\!60}a^{9}-\frac{10\!\cdots\!03}{24\!\cdots\!65}a^{8}+\frac{81\!\cdots\!89}{97\!\cdots\!60}a^{7}-\frac{47\!\cdots\!11}{48\!\cdots\!30}a^{6}+\frac{67\!\cdots\!43}{97\!\cdots\!60}a^{5}+\frac{19\!\cdots\!21}{97\!\cdots\!60}a^{4}-\frac{18\!\cdots\!19}{97\!\cdots\!66}a^{3}+\frac{24\!\cdots\!29}{97\!\cdots\!60}a^{2}-\frac{29\!\cdots\!16}{24\!\cdots\!65}a-\frac{14\!\cdots\!77}{33\!\cdots\!54}$, $\frac{12\!\cdots\!93}{16\!\cdots\!50}a^{15}-\frac{20\!\cdots\!33}{10\!\cdots\!00}a^{14}+\frac{94\!\cdots\!56}{25\!\cdots\!25}a^{13}-\frac{88\!\cdots\!39}{10\!\cdots\!00}a^{12}+\frac{65\!\cdots\!89}{10\!\cdots\!00}a^{11}-\frac{38\!\cdots\!31}{25\!\cdots\!25}a^{10}+\frac{25\!\cdots\!09}{50\!\cdots\!45}a^{9}-\frac{34\!\cdots\!99}{34\!\cdots\!00}a^{8}+\frac{17\!\cdots\!59}{10\!\cdots\!90}a^{7}-\frac{55\!\cdots\!81}{33\!\cdots\!00}a^{6}+\frac{44\!\cdots\!58}{83\!\cdots\!75}a^{5}-\frac{15\!\cdots\!43}{50\!\cdots\!45}a^{4}-\frac{31\!\cdots\!91}{50\!\cdots\!50}a^{3}-\frac{81\!\cdots\!98}{16\!\cdots\!15}a^{2}-\frac{55\!\cdots\!47}{10\!\cdots\!00}a-\frac{92\!\cdots\!01}{11\!\cdots\!00}$, $\frac{87\!\cdots\!59}{71\!\cdots\!00}a^{15}-\frac{38\!\cdots\!59}{71\!\cdots\!00}a^{14}+\frac{12\!\cdots\!23}{17\!\cdots\!75}a^{13}-\frac{89\!\cdots\!31}{35\!\cdots\!50}a^{12}+\frac{33\!\cdots\!19}{23\!\cdots\!00}a^{11}-\frac{76\!\cdots\!63}{17\!\cdots\!75}a^{10}+\frac{48\!\cdots\!24}{35\!\cdots\!95}a^{9}-\frac{10\!\cdots\!89}{35\!\cdots\!50}a^{8}+\frac{40\!\cdots\!61}{71\!\cdots\!90}a^{7}-\frac{31\!\cdots\!97}{59\!\cdots\!25}a^{6}+\frac{90\!\cdots\!83}{71\!\cdots\!00}a^{5}+\frac{31\!\cdots\!73}{47\!\cdots\!60}a^{4}-\frac{55\!\cdots\!03}{35\!\cdots\!50}a^{3}-\frac{15\!\cdots\!78}{35\!\cdots\!95}a^{2}+\frac{53\!\cdots\!19}{71\!\cdots\!00}a-\frac{42\!\cdots\!69}{39\!\cdots\!50}$, $\frac{33\!\cdots\!11}{47\!\cdots\!60}a^{15}-\frac{11\!\cdots\!99}{23\!\cdots\!30}a^{14}+\frac{18\!\cdots\!29}{47\!\cdots\!60}a^{13}-\frac{26\!\cdots\!49}{11\!\cdots\!65}a^{12}+\frac{31\!\cdots\!47}{47\!\cdots\!60}a^{11}-\frac{15\!\cdots\!77}{47\!\cdots\!60}a^{10}+\frac{20\!\cdots\!91}{47\!\cdots\!26}a^{9}-\frac{34\!\cdots\!31}{23\!\cdots\!30}a^{8}-\frac{70\!\cdots\!17}{11\!\cdots\!65}a^{7}+\frac{18\!\cdots\!35}{47\!\cdots\!26}a^{6}-\frac{21\!\cdots\!57}{95\!\cdots\!52}a^{5}+\frac{30\!\cdots\!19}{11\!\cdots\!65}a^{4}-\frac{12\!\cdots\!39}{47\!\cdots\!60}a^{3}+\frac{16\!\cdots\!98}{11\!\cdots\!65}a^{2}-\frac{56\!\cdots\!21}{47\!\cdots\!60}a+\frac{19\!\cdots\!57}{52\!\cdots\!40}$, $\frac{37\!\cdots\!39}{59\!\cdots\!25}a^{15}-\frac{14\!\cdots\!13}{71\!\cdots\!00}a^{14}+\frac{11\!\cdots\!07}{35\!\cdots\!50}a^{13}-\frac{15\!\cdots\!51}{17\!\cdots\!75}a^{12}+\frac{21\!\cdots\!27}{35\!\cdots\!50}a^{11}-\frac{10\!\cdots\!09}{71\!\cdots\!00}a^{10}+\frac{13\!\cdots\!49}{28\!\cdots\!56}a^{9}-\frac{60\!\cdots\!41}{71\!\cdots\!00}a^{8}+\frac{23\!\cdots\!43}{14\!\cdots\!80}a^{7}-\frac{12\!\cdots\!01}{23\!\cdots\!00}a^{6}-\frac{10\!\cdots\!93}{23\!\cdots\!00}a^{5}+\frac{13\!\cdots\!03}{71\!\cdots\!90}a^{4}+\frac{18\!\cdots\!33}{71\!\cdots\!00}a^{3}-\frac{69\!\cdots\!07}{47\!\cdots\!60}a^{2}-\frac{35\!\cdots\!37}{71\!\cdots\!00}a-\frac{39\!\cdots\!53}{39\!\cdots\!50}$, $\frac{21\!\cdots\!47}{23\!\cdots\!00}a^{15}+\frac{22\!\cdots\!87}{59\!\cdots\!25}a^{14}+\frac{18\!\cdots\!29}{59\!\cdots\!25}a^{13}+\frac{50\!\cdots\!09}{23\!\cdots\!00}a^{12}+\frac{87\!\cdots\!24}{59\!\cdots\!25}a^{11}+\frac{43\!\cdots\!57}{11\!\cdots\!50}a^{10}-\frac{11\!\cdots\!21}{23\!\cdots\!30}a^{9}+\frac{68\!\cdots\!11}{23\!\cdots\!00}a^{8}-\frac{21\!\cdots\!73}{47\!\cdots\!26}a^{7}+\frac{25\!\cdots\!93}{23\!\cdots\!00}a^{6}-\frac{51\!\cdots\!31}{23\!\cdots\!00}a^{5}-\frac{34\!\cdots\!97}{47\!\cdots\!60}a^{4}+\frac{46\!\cdots\!71}{11\!\cdots\!50}a^{3}+\frac{39\!\cdots\!47}{47\!\cdots\!26}a^{2}-\frac{51\!\cdots\!42}{59\!\cdots\!25}a-\frac{66\!\cdots\!99}{26\!\cdots\!00}$, $\frac{83\!\cdots\!27}{11\!\cdots\!50}a^{15}-\frac{93\!\cdots\!23}{24\!\cdots\!00}a^{14}+\frac{28\!\cdots\!31}{71\!\cdots\!00}a^{13}-\frac{10\!\cdots\!71}{71\!\cdots\!00}a^{12}+\frac{49\!\cdots\!41}{71\!\cdots\!00}a^{11}-\frac{33\!\cdots\!69}{17\!\cdots\!75}a^{10}+\frac{68\!\cdots\!81}{14\!\cdots\!80}a^{9}-\frac{49\!\cdots\!34}{61\!\cdots\!75}a^{8}+\frac{16\!\cdots\!33}{28\!\cdots\!56}a^{7}+\frac{43\!\cdots\!44}{59\!\cdots\!25}a^{6}-\frac{13\!\cdots\!13}{82\!\cdots\!00}a^{5}-\frac{15\!\cdots\!63}{14\!\cdots\!80}a^{4}+\frac{42\!\cdots\!08}{17\!\cdots\!75}a^{3}+\frac{10\!\cdots\!39}{47\!\cdots\!60}a^{2}+\frac{73\!\cdots\!11}{35\!\cdots\!50}a+\frac{73\!\cdots\!64}{19\!\cdots\!25}$ Copy content Toggle raw display (assuming GRH)
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:  \( 14811624.1093 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 14811624.1093 \cdot 1}{2\cdot\sqrt{2106797133618518949609765625}}\cr\approx \mathstrut & 0.391922569496 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869)
 
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^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869, 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^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869);
 
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^16 - 4*x^15 + 56*x^14 - 182*x^13 + 1092*x^12 - 3072*x^11 + 9994*x^10 - 20362*x^9 + 40781*x^8 - 34526*x^7 + 13090*x^6 + 38884*x^5 + 15301*x^4 - 14428*x^3 + 40421*x^2 + 9424*x + 27869);
 
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

$C_2^3:C_4$ (as 16T33):

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 32
The 11 conjugacy class representatives for $C_2^3:C_4$
Character table for $C_2^3:C_4$

Intermediate fields

\(\Q(\sqrt{745}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{5}) \), 4.0.3725.1 x2, \(\Q(\sqrt{5}, \sqrt{149})\), 4.0.111005.1 x2, 8.0.308052750625.2 x2, 8.4.45899859843125.1 x2, 8.0.308052750625.1

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

Galois closure: deg 32
Degree 8 siblings: 8.4.45899859843125.1, 8.0.273563164665025.1, 8.4.6839079116625625.1, 8.0.308052750625.2
Degree 16 siblings: 16.0.1870920126538589568011456265625.3, 16.8.46773003163464739200286406640625.1, 16.0.46773003163464739200286406640625.3
Minimal sibling: 8.0.308052750625.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{4}$ ${\href{/padicField/3.4.0.1}{4} }^{4}$ R ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.2.0.1}{2} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{4}$ ${\href{/padicField/29.1.0.1}{1} }^{16}$ ${\href{/padicField/31.2.0.1}{2} }^{8}$ ${\href{/padicField/37.2.0.1}{2} }^{8}$ ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.4.0.1}{4} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{4}$

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
\(5\) Copy content Toggle raw display 5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(149\) Copy content Toggle raw display 149.4.2.1$x^{4} + 26514 x^{3} + 177649591 x^{2} + 25208742294 x + 634090662$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
149.4.3.1$x^{4} + 596$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.3.1$x^{4} + 596$$4$$1$$3$$C_4$$[\ ]_{4}$
149.4.2.1$x^{4} + 26514 x^{3} + 177649591 x^{2} + 25208742294 x + 634090662$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$