Properties

Label 20070p1
Conductor 2007020070
Discriminant 2.540×10122.540\times 10^{12}
j-invariant 6407045708273293484375000 \frac{640704570827329}{3484375000}
CM no
Rank 11
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy=x3x216164x+791320y^2+xy=x^3-x^2-16164x+791320 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz=x3x2z16164xz2+791320z3y^2z+xyz=x^3-x^2z-16164xz^2+791320z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x3258627x+50385854y^2=x^3-258627x+50385854 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, -1, 0, -16164, 791320])
 
gp: E = ellinit([1, -1, 0, -16164, 791320])
 
magma: E := EllipticCurve([1, -1, 0, -16164, 791320]);
 
oscar: E = elliptic_curve([1, -1, 0, -16164, 791320])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(51,287)(51, 287)0.612709996826479938500875189940.61270999682647993850087518994\infty

Integral points

(51,287) \left(51, 287\right) , (51,338) \left(51, -338\right) , (81,52) \left(81, 52\right) , (81,133) \left(81, -133\right) Copy content Toggle raw display

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

Invariants

Conductor: NN  =  20070 20070  = 23252232 \cdot 3^{2} \cdot 5 \cdot 223
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  25401093750002540109375000 = 2336592232^{3} \cdot 3^{6} \cdot 5^{9} \cdot 223
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  6407045708273293484375000 \frac{640704570827329}{3484375000}  = 235922318620932^{-3} \cdot 5^{-9} \cdot 223^{-1} \cdot 86209^{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}} ≈ 1.22382770880212184463985979511.2238277088021218446398597951
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 0.674521564468066998942237176640.67452156446806699894223717664
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.91612560662130030.9161256066213003
Szpiro ratio: σm\sigma_{m} ≈ 4.1067265089495194.106726508949519

BSD invariants

Analytic rank: ranr_{\mathrm{an}} = 1 1
sage: E.analytic_rank()
 
gp: ellanalyticrank(E)
 
magma: AnalyticRank(E);
 
Mordell-Weil rank: rr = 1 1
comment: Rank
 
sage: E.rank()
 
gp: [lower,upper] = ellrank(E)
 
magma: Rank(E);
 
Regulator: Reg(E/Q)\mathrm{Reg}(E/\Q) ≈ 0.612709996826479938500875189940.61270999682647993850087518994
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 0.816870973066983020293591548660.81687097306698302029359154866
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 = 9 9  = 11321 1\cdot1\cdot3^{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}} = 11
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) ≈ 4.50454510183963271111734137174.5045451018396327111173413717
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    (rounded)
comment: Order of Sha
 
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 

BSD formula

4.504545102L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.8168710.6127109124.504545102\displaystyle 4.504545102 \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 0.816871 \cdot 0.612710 \cdot 9}{1^2} \approx 4.504545102

# 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   20070.2.a.r

qq2+q4+q5+3q7q8q10+2q112q133q14+q164q17q19+O(q20) q - q^{2} + q^{4} + q^{5} + 3 q^{7} - q^{8} - q^{10} + 2 q^{11} - 2 q^{13} - 3 q^{14} + q^{16} - 4 q^{17} - 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: 45360
comment: Modular degree
 
sage: E.modular_degree()
 
gp: ellmoddegree(E)
 
magma: ModularDegree(E);
 
Γ0(N) \Gamma_0(N) -optimal: yes
Manin constant: 1
comment: Manin constant
 
magma: ManinConstant(E);
 

Local data at primes of bad reduction

This elliptic curve is not semistable. There are 4 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 11 I3I_{3} nonsplit multiplicative 1 1 3 3
33 11 I0I_0^{*} additive -1 2 6 0
55 99 I9I_{9} split multiplicative -1 1 9 9
223223 11 I1I_{1} nonsplit 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.

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 = [[2231, 2, 2231, 3], [1, 0, 2, 1], [1, 2, 0, 1], [8919, 2, 8918, 3], [1, 1, 8919, 0], [7137, 2, 7137, 3], [4461, 2, 4461, 3], [5801, 2, 5801, 3]]
 
GL(2,Integers(8920)).subgroup(gens)
 
Gens := [[2231, 2, 2231, 3], [1, 0, 2, 1], [1, 2, 0, 1], [8919, 2, 8918, 3], [1, 1, 8919, 0], [7137, 2, 7137, 3], [4461, 2, 4461, 3], [5801, 2, 5801, 3]];
 
sub<GL(2,Integers(8920))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 8920=235223 8920 = 2^{3} \cdot 5 \cdot 223 , index 22, genus 00, and generators

(2231222313),(1021),(1201),(8919289183),(1189190),(7137271373),(4461244613),(5801258013)\left(\begin{array}{rr} 2231 & 2 \\ 2231 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8919 & 2 \\ 8918 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 8919 & 0 \end{array}\right),\left(\begin{array}{rr} 7137 & 2 \\ 7137 & 3 \end{array}\right),\left(\begin{array}{rr} 4461 & 2 \\ 4461 & 3 \end{array}\right),\left(\begin{array}{rr} 5801 & 2 \\ 5801 & 3 \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[8920])K:=\Q(E[8920]) is a degree-907530621419520907530621419520 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/8920Z)\GL_2(\Z/8920\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 nonsplit multiplicative 44 10035=325223 10035 = 3^{2} \cdot 5 \cdot 223
33 additive 66 223 223
55 split multiplicative 66 4014=232223 4014 = 2 \cdot 3^{2} \cdot 223
223223 nonsplit multiplicative 224224 90=2325 90 = 2 \cdot 3^{2} \cdot 5

Isogenies

gp: ellisomat(E)
 

This curve has no rational isogenies. Its isogeny class 20070p consists of this curve only.

Twists

The minimal quadratic twist of this elliptic curve is 2230a1, its twist by 3-3.

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} (which is trivial) are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
33 3.3.8920.1 Z/2Z\Z/2\Z not in database
66 6.6.709732288000.1 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
88 8.2.5408392915467.2 Z/3Z\Z/3\Z not in database
1212 deg 12 Z/4Z\Z/4\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 3 5 7 11 13 17 19 23 29 31 37 41 43 47 223
Reduction type nonsplit add split ord ord ord ord ord ord ord ord ord ord ord ord nonsplit
λ\lambda-invariant(s) 3 - 2 3 1 1 1 1 1 1 3 1 1 3 1 1
μ\mu-invariant(s) 0 - 0 0 0 0 0 0 0 0 0 0 0 0 0 0

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

pp-adic regulators

Note: pp-adic regulator data only exists for primes p5p\ge 5 of good ordinary reduction.