Properties

Label 5010h1
Conductor 50105010
Discriminant 3.652×1012-3.652\times 10^{12}
j-invariant 3585314011219213652290000000 -\frac{358531401121921}{3652290000000}
CM no
Rank 00
Torsion structure Z/7Z\Z/{7}\Z

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2+xy=x31480x+94400y^2+xy=x^3-1480x+94400 Copy content Toggle raw display (homogenize, simplify)
y2z+xyz=x31480xz2+94400z3y^2z+xyz=x^3-1480xz^2+94400z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x31918107x+4410080694y^2=x^3-1918107x+4410080694 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([1, 0, 0, -1480, 94400])
 
gp: E = ellinit([1, 0, 0, -1480, 94400])
 
magma: E := EllipticCurve([1, 0, 0, -1480, 94400]);
 
oscar: E = elliptic_curve([1, 0, 0, -1480, 94400])
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 
oscar: short_weierstrass_model(E)
 

Mordell-Weil group structure

Z/7Z\Z/{7}\Z

magma: MordellWeilGroup(E);
 

Mordell-Weil generators

PPh^(P)\hat{h}(P)Order
(20,260)(20, 260)0077

Integral points

(40,320) \left(-40, 320\right) , (40,280) \left(-40, -280\right) , (20,260) \left(20, 260\right) , (20,280) \left(20, -280\right) , (110,1070) \left(110, 1070\right) , (110,1180) \left(110, -1180\right) Copy content Toggle raw display

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

Invariants

Conductor: NN  =  5010 5010  = 2351672 \cdot 3 \cdot 5 \cdot 167
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  3652290000000-3652290000000 = 1273757167-1 \cdot 2^{7} \cdot 3^{7} \cdot 5^{7} \cdot 167
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  3585314011219213652290000000 -\frac{358531401121921}{3652290000000}  = 1273757193167137393-1 \cdot 2^{-7} \cdot 3^{-7} \cdot 5^{-7} \cdot 19^{3} \cdot 167^{-1} \cdot 3739^{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}} ≈ 1.09273794400213222130726492381.0927379440021322213072649238
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.09273794400213222130726492381.0927379440021322213072649238
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 0.96116877240030370.9611687724003037
Szpiro ratio: σm\sigma_{m} ≈ 4.2769728634086394.276972863408639

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 ≈ 0.672064285556499615360381569130.67206428555649961536038156913
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 = 343 343  = 7771 7\cdot7\cdot7\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}} = 77
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) ≈ 4.70444999889549730752267098394.7044499988954973075226709839
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

4.704449999L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor210.6720641.000000343724.704449999\displaystyle 4.704449999 \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 0.672064 \cdot 1.000000 \cdot 343}{7^2} \approx 4.704449999

# 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   5010.2.a.h

q+q2+q3+q4+q5+q6+q7+q8+q9+q102q11+q12+q14+q15+q16+4q17+q18q19+O(q20) q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 2 q^{11} + q^{12} + q^{14} + q^{15} + q^{16} + 4 q^{17} + q^{18} - 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: 9408
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 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 77 I7I_{7} split multiplicative -1 1 7 7
33 77 I7I_{7} split multiplicative -1 1 7 7
55 77 I7I_{7} split multiplicative -1 1 7 7
167167 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 except those listed in the table below.

prime \ell mod-\ell image \ell-adic image
77 7B.1.1 7.48.0.1

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 = [[8, 7, 35063, 140274], [117608, 7, 107513, 140274], [8, 7, 70133, 140274], [1, 14, 0, 1], [46768, 7, 46753, 140274], [105211, 70154, 0, 65131], [84176, 7, 84161, 140274], [1, 0, 14, 1], [8, 5, 91, 57], [140267, 14, 140266, 15]]
 
GL(2,Integers(140280)).subgroup(gens)
 
Gens := [[8, 7, 35063, 140274], [117608, 7, 107513, 140274], [8, 7, 70133, 140274], [1, 14, 0, 1], [46768, 7, 46753, 140274], [105211, 70154, 0, 65131], [84176, 7, 84161, 140274], [1, 0, 14, 1], [8, 5, 91, 57], [140267, 14, 140266, 15]];
 
sub<GL(2,Integers(140280))|Gens>;
 

The image H:=ρE(Gal(Q/Q))H:=\rho_E(\Gal(\overline{\Q}/\Q)) of the adelic Galois representation has level 140280=23357167 140280 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 167 , index 9696, genus 22, and generators

(8735063140274),(1176087107513140274),(8770133140274),(11401),(46768746753140274),(10521170154065131),(84176784161140274),(10141),(859157),(1402671414026615)\left(\begin{array}{rr} 8 & 7 \\ 35063 & 140274 \end{array}\right),\left(\begin{array}{rr} 117608 & 7 \\ 107513 & 140274 \end{array}\right),\left(\begin{array}{rr} 8 & 7 \\ 70133 & 140274 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 46768 & 7 \\ 46753 & 140274 \end{array}\right),\left(\begin{array}{rr} 105211 & 70154 \\ 0 & 65131 \end{array}\right),\left(\begin{array}{rr} 84176 & 7 \\ 84161 & 140274 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 140267 & 14 \\ 140266 & 15 \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[140280])K:=\Q(E[140280]) is a degree-574559373376880640574559373376880640 Galois extension of Q\Q with Gal(K/Q)\Gal(K/\Q) isomorphic to the projection of HH to GL2(Z/140280Z)\GL_2(\Z/140280\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 split multiplicative 44 2505=35167 2505 = 3 \cdot 5 \cdot 167
33 split multiplicative 44 1670=25167 1670 = 2 \cdot 5 \cdot 167
55 split multiplicative 66 1002=23167 1002 = 2 \cdot 3 \cdot 167
77 good 22 167 167
167167 nonsplit multiplicative 168168 30=235 30 = 2 \cdot 3 \cdot 5

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 7.
Its isogeny class 5010h consists of 2 curves linked by isogenies of degree 7.

Twists

This elliptic curve is its own minimal quadratic twist.

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/7Z\cong \Z/{7}\Z are as follows:

[K:Q][K:\Q] KK E(K)torsE(K)_{\rm tors} Base change curve
33 3.1.20040.1 Z/14Z\Z/14\Z not in database
66 6.0.8048096064000.1 Z/2ZZ/14Z\Z/2\Z \oplus \Z/14\Z not in database
88 deg 8 Z/21Z\Z/21\Z not in database
1212 deg 12 Z/28Z\Z/28\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 7 167
Reduction type split split split ord nonsplit
λ\lambda-invariant(s) 2 1 1 4 0
μ\mu-invariant(s) 0 0 0 0 0

All Iwasawa λ\lambda and μ\mu-invariants for primes p11p\ge 11 of good reduction are zero.

pp-adic regulators

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