Properties

Label 10800.a1
Conductor 1080010800
Discriminant 270000-270000
j-invariant 0 0
CM yes (D=3D=-3)
Rank 11
Torsion structure trivial

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Show commands: Magma / Oscar / PariGP / SageMath

Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

y2=x325y^2=x^3-25 Copy content Toggle raw display (homogenize, simplify)
y2z=x325z3y^2z=x^3-25z^3 Copy content Toggle raw display (dehomogenize, simplify)
y2=x325y^2=x^3-25 Copy content Toggle raw display (homogenize, minimize)

comment: Define the curve
 
sage: E = EllipticCurve([0, 0, 0, 0, -25])
 
gp: E = ellinit([0, 0, 0, 0, -25])
 
magma: E := EllipticCurve([0, 0, 0, 0, -25]);
 
oscar: E = elliptic_curve([0, 0, 0, 0, -25])
 
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
(5,10)(5, 10)0.878596604659648864944941946730.87859660465964886494494194673\infty

Integral points

(5,±10)(5,\pm 10) Copy content Toggle raw display

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

Invariants

Conductor: NN  =  10800 10800  = 2433522^{4} \cdot 3^{3} \cdot 5^{2}
comment: Conductor
 
sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
oscar: conductor(E)
 
Discriminant: Δ\Delta  =  270000-270000 = 1243354-1 \cdot 2^{4} \cdot 3^{3} \cdot 5^{4}
comment: Discriminant
 
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
oscar: discriminant(E)
 
j-invariant: jj  =  0 0  = 00
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[(1+3)/2]\Z[(1+\sqrt{-3})/2]    (potential complex multiplication)
sage: E.has_cm()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: ST(E)\mathrm{ST}(E) = N(U(1))N(\mathrm{U}(1))
Faltings height: hFaltingsh_{\mathrm{Faltings}} ≈ 0.27893599192966193080172740173-0.27893599192966193080172740173
gp: ellheight(E)
 
magma: FaltingsHeight(E);
 
oscar: faltings_height(E)
 
Stable Faltings height: hstableh_{\mathrm{stable}} ≈ 1.3211174284280379149898691959-1.3211174284280379149898691959
magma: StableFaltingsHeight(E);
 
oscar: stable_faltings_height(E)
 
abcabc quality: QQ ≈ 
Szpiro ratio: σm\sigma_{m} ≈ 2.14926772583935352.1492677258393535

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) ≈ 0.878596604659648864944941946730.87859660465964886494494194673
comment: Regulator
 
sage: E.regulator()
 
G = E.gen \\ if available
 
matdet(ellheightmatrix(E,G))
 
magma: Regulator(E);
 
Real period: Ω\Omega ≈ 1.42028351486909163415809589471.4202835148690916341580958947
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 = 3 3  = 113 1\cdot1\cdot3
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) ≈ 3.74356882145416746845856698053.7435688214541674684585669805
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

3.743568821L(E,1)=#Ш(E/Q)ΩEReg(E/Q)pcp#E(Q)tor211.4202840.8785973123.743568821\displaystyle 3.743568821 \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 1.420284 \cdot 0.878597 \cdot 3}{1^2} \approx 3.743568821

# 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   10800.2.a.a

q5q7+2q13+q19+O(q20) q - 5 q^{7} + 2 q^{13} + 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: 1296
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 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 11 IIII additive -1 4 4 0
33 11 IIII additive 1 3 3 0
55 33 IVIV additive -1 2 4 0

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)];
 

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 22 675=3352 675 = 3^{3} \cdot 5^{2}
33 additive 22 400=2452 400 = 2^{4} \cdot 5^{2}
55 additive 1414 432=2433 432 = 2^{4} \cdot 3^{3}

Isogenies

gp: ellisomat(E)
 

This curve has non-trivial cyclic isogenies of degree dd for d=d= 3.
Its isogeny class 10800.a consists of 2 curves linked by isogenies of degree 3.

Twists

The minimal quadratic twist of this elliptic curve is 2700.u2, its twist by 4-4.

The minimal sextic twist of this elliptic curve is 27.a4, its sextic twist by 10000-10000.

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
22 Q(1)\Q(\sqrt{-1}) Z/3Z\Z/3\Z not in database
33 3.1.675.1 Z/2Z\Z/2\Z not in database
66 6.0.1366875.1 Z/2ZZ/2Z\Z/2\Z \oplus \Z/2\Z not in database
66 6.2.4320000.2 Z/3Z\Z/3\Z not in database
66 6.0.29160000.1 Z/6Z\Z/6\Z not in database
1212 deg 12 Z/4Z\Z/4\Z not in database
1212 12.0.18662400000000.1 Z/3ZZ/3Z\Z/3\Z \oplus \Z/3\Z not in database
1212 deg 12 Z/7Z\Z/7\Z not in database
1212 12.0.7652750400000000.2 Z/2ZZ/6Z\Z/2\Z \oplus \Z/6\Z not in database
1818 18.0.10411482432835584000000000000.3 Z/9Z\Z/9\Z not in database
1818 18.2.42845606719488000000000000.1 Z/6Z\Z/6\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
Reduction type add add add ord ss ord ss ord ss ss ord ord ss ord ss
λ\lambda-invariant(s) - - - 1 1,1 1 1,1 3 1,1 1,1 1 1 1,1 1 1,1
μ\mu-invariant(s) - - - 0 0,0 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.