Properties

Label 4160o3
Conductor 41604160
Discriminant 10649600001064960000
j-invariant 96364915388125 \frac{9636491538}{8125}
CM no
Rank 00
Torsion structure Z/4Z\Z/{4}\Z

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x32252x+41104y^2=x^3-2252x+41104 Copy content Toggle raw display (homogenize, simplify)
y2z=x32252xz2+41104z3y^2z=x^3-2252xz^2+41104z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x32252x+41104y^2=x^3-2252x+41104 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 0, -2252, 41104])
 
gp: E = ellinit([0, 0, 0, -2252, 41104])
 
magma: E := EllipticCurve([0, 0, 0, -2252, 41104]);
 
oscar: E = elliptic_curve([0, 0, 0, -2252, 41104])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z/4Z\Z/{4}\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(18,80)(18, 80)0044

Integral points

(18,±80)(18,\pm 80), (28,0) \left(28, 0\right) Copy content Toggle raw display

comment: Integral points
 
sage: E.integral_points()
 
magma: IntegralPoints(E);
 

Invariants

Conductor: NN  =  4160 4160  = 265132^{6} \cdot 5 \cdot 13
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  10649600001064960000 = 21754132^{17} \cdot 5^{4} \cdot 13
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  96364915388125 \frac{9636491538}{8125}  = 2335413156332 \cdot 3^{3} \cdot 5^{-4} \cdot 13^{-1} \cdot 563^{3}
comment: j-invariant
 
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
oscar: j_invariant(E)
 
Endomorphism ring: End(E)\mathrm{End}(E) = Z\Z
Geometric endomorphism ring: End(EQ)\mathrm{End}(E_{\overline{\Q}})  =  Z\Z    (no potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = SU(2)\mathrm{SU}(2)
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 0.659694265922581178785213741380.65969426592258117878521374138
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.32226423987067467622253176402-0.32226423987067467622253176402
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.94288736503599630.9428873650359963
Szpiro ratio: σm\sigma_{m} ≈ 4.17271053577650354.1727105357765035

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 0 0
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 0 0
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) = 11
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 1.54268548069840645428454257791.5426854806984064542845425779
comment: Real Period
 
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Tamagawa product: pcp\prod_{p}c_p = 16 16  = 22221 2^{2}\cdot2^{2}\cdot1
comment: Tamagawa numbers
 
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
oscar: tamagawa_numbers(E)
 
Torsion order: #E(Q)tor\#E(\Q)_{\mathrm{tor}} = 44
comment: Torsion order
 
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
oscar: prod(torsion_structure(E)[1])
 
Special value: L(E,1) L(E,1) ≈ 1.54268548069840645428454257791.5426854806984064542845425779
comment: Special L-value
 
r = E.rank();
 
E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Analytic order of Ш: Шan{}_{\mathrm{an}}  =  11    (exact)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

1.542685481L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor211.5426851.00000016421.542685481\displaystyle 1.542685481 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 1.542685 \cdot 1.000000 \cdot 16}{4^2} \approx 1.542685481

# self-contained SageMath code snippet for the BSD formula (checks rank, computes analytic sha)
 
E = EllipticCurve(%s); r = E.rank(); ar = E.analytic_rank(); assert r == ar;
 
Lr1 = E.lseries().dokchitser().derivative(1,r)/r.factorial(); sha = E.sha().an_numerical();
 
omega = E.period_lattice().omega(); reg = E.regulator(); tam = E.tamagawa_product(); tor = E.torsion_order();
 
assert r == ar; print("analytic sha: " + str(RR(Lr1) * tor^2 / (omega * reg * tam)))
 
/* self-contained Magma code snippet for the BSD formula (checks rank, computes analytic sha) */
 
E := EllipticCurve(%s); r := Rank(E); ar,Lr1 := AnalyticRank(E: Precision := 12); assert r eq ar;
 
sha := MordellWeilShaInformation(E); omega := RealPeriod(E) * (Discriminant(E) gt 0 select 2 else 1);
 
reg := Regulator(E); tam := &*TamagawaNumbers(E); tor := #TorsionSubgroup(E);
 
assert r eq ar; print "analytic sha:", Lr1 * tor^2 / (omega * reg * tam);
 

Modular invariants

Modular form   4160.2.a.l

q+q53q94q11+q136q17+4q19+O(q20) q + q^{5} - 3 q^{9} - 4 q^{11} + q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20}) Copy content Toggle raw display

comment: q-expansion of modular form
 
sage: E.q_eigenform(20)
 
\\ actual modular form, use for small N
 
[mf,F] = mffromell(E)
 
Ser(mfcoefs(mf,20),q)
 
\\ or just the series
 
Ser(ellan(E,20),q)*q
 
magma: ModularForm(E);
 

For more coefficients, see the Downloads section to the right.

Modular degree: 2048
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: no
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 3 primes pp of bad reduction:

pp Tamagawa number Kodaira symbol Reduction type Root number ordp(N)\mathrm{ord}_p(N) ordp(Δ)\mathrm{ord}_p(\Delta) ordp(den(j))\mathrm{ord}_p(\mathrm{den}(j))
22 44 I7I_{7}^{*} additive -1 6 17 0
55 44 I4I_{4} split multiplicative -1 1 4 4
1313 11 I1I_{1} split multiplicative -1 1 1 1

comment: Local data
 
sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
oscar: [(p,tamagawa_number(E,p), kodaira_symbol(E,p), reduction_type(E,p)) for p in bad_primes(E)]
 

Galois representations

The \ell-adic Galois representation has maximal image for all primes \ell except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
22 2B 4.12.0.7

comment: mod p Galois image
 
sage: rho = E.galois_representation(); [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

gens = [[513, 8, 512, 9], [1, 0, 8, 1], [192, 447, 333, 346], [1, 8, 0, 1], [328, 3, 205, 2], [1, 4, 4, 17], [417, 8, 108, 33], [7, 6, 514, 515], [317, 324, 278, 59]]
 
GL(2,Integers(520)).subgroup(gens)
 
Gens := [[513, 8, 512, 9], [1, 0, 8, 1], [192, 447, 333, 346], [1, 8, 0, 1], [328, 3, 205, 2], [1, 4, 4, 17], [417, 8, 108, 33], [7, 6, 514, 515], [317, 324, 278, 59]];
 
sub<GL(2,Integers(520))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 520=23513 520 = 2^{3} \cdot 5 \cdot 13 , index 4848, genus 00, and generators

(51385129),(1081),(192447333346),(1801),(32832052),(14417),(417810833),(76514515),(31732427859)\left(\begin{array}{rr} 513 & 8 \\ 512 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 192 & 447 \\ 333 & 346 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 328 & 3 \\ 205 & 2 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 417 & 8 \\ 108 & 33 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 514 & 515 \end{array}\right),\left(\begin{array}{rr} 317 & 324 \\ 278 & 59 \end{array}\right).

Input positive integer mm to see the generators of the reduction of HH to GL2(Z/mZ)\mathrm{GL}_2(\Z/m\Z):

The torsion field K:=Q(E[520])K:=\Q(E[520]) is a degree-402554880402554880 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/520Z)\GL_2(\Z/520\Z).

The table below list all primes \ell for which the Serre invariants associated to the mod-\ell Galois representation are exceptional.

\ell Reduction type Serre weight Serre conductor
22 additive 44 13 13
55 split multiplicative 66 832=2613 832 = 2^{6} \cdot 13
1313 split multiplicative 1414 320=265 320 = 2^{6} \cdot 5

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 2 and 4.
Its isogeny class 4160o consists of 4 curves linked by isogenies of degrees dividing 4.

Twists

The minimal quadratic twist of this elliptic curve is 520a3, its twist by 8-8.

Growth of torsion in number fields

The number fields KK of degree less than 24 such that E(K)torsE(K)_{\rm tors} is strictly larger than E(Q)torsE(\Q)_{\rm tors} Z/4Z\cong \Z/{4}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
22 Q(26)\Q(\sqrt{26}) Z/2ZZ/4Z\Z/2\Z \oplus \Z/4\Z not in database
44 4.0.665600.1 Z/8Z\Z/8\Z not in database
88 8.0.80980417183744.23 Z/4ZZ/4Z\Z/4\Z \oplus \Z/4\Z not in database
88 8.8.197706096640000.8 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 8.0.299483791360000.28 Z/2ZZ/8Z\Z/2\Z \oplus \Z/8\Z not in database
88 deg 8 Z/12Z\Z/12\Z not in database
1616 deg 16 Z/16Z\Z/16\Z not in database
1616 deg 16 Z/2ZZ/12Z\Z/2\Z \oplus \Z/12\Z not in database

We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.

Iwasawa invariants

pp 2 5 13
Reduction type add split split
λ\lambda-invariant(s) - 3 1
μ\mu-invariant(s) - 0 0

All Iwasawa λ\lambda and μ\mu-invariants for primes p3p\ge 3 of good reduction are zero.

An entry - indicates that the invariants are not computed because the reduction is additive.

pp-adic regulators

All pp-adic regulators are identically 11 since the rank is 00.