Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-15200x-722000\)
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(homogenize, simplify) |
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\(y^2z=x^3-15200xz^2-722000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15200x-722000\)
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(homogenize, minimize) |
comment: Define the curve
sage: E = EllipticCurve([0, 0, 0, -15200, -722000])
gp: E = ellinit([0, 0, 0, -15200, -722000])
magma: E := EllipticCurve([0, 0, 0, -15200, -722000]);
oscar: E = elliptic_curve([0, 0, 0, -15200, -722000])
sage: E.short_weierstrass_model()
magma: WeierstrassModel(E);
oscar: short_weierstrass_model(E)
Mordell-Weil group structure
trivial
magma: MordellWeilGroup(E);
Invariants
| Conductor: | $N$ | = | \( 144400 \) | = | $2^{4} \cdot 5^{2} \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $\Delta$ | = | $-438976000000$ | = | $-1 \cdot 2^{12} \cdot 5^{6} \cdot 19^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | $j$ | = | \( -884736 \) | = | $-1 \cdot 2^{15} \cdot 3^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z[(1+\sqrt{-19})/2]\) (potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $N(\mathrm{U}(1))$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.1477174768862206733133363500$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0862584046823849384065322960$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $Q$ | ≈ | $1.3175706029138485$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.409220429024579$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Mordell-Weil rank: | $r$ | = | $ 0$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $\Omega$ | ≈ | $0.21497068018254422901565002995$ | 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);
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot1\cdot2 $ | 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)
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Special value: | $ L(E,1)$ | ≈ | $3.8694722432857961222817005391 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 3.869472243 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{9 \cdot 0.214971 \cdot 1.000000 \cdot 2}{1^2} \approx 3.869472243$
# 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 144400.2.a.bs
\( q + 3 q^{7} - 3 q^{9} + 5 q^{11} + 7 q^{17} + O(q^{20}) \)
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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: | 201600 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $II^{*}$ | additive | -1 | 4 | 12 | 0 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $2$ | $III$ | additive | 1 | 2 | 3 | 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 |
|---|---|---|---|
| $2$ | additive | $2$ | \( 475 = 5^{2} \cdot 19 \) |
| $5$ | additive | $14$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $110$ | \( 400 = 2^{4} \cdot 5^{2} \) |
Isogenies
gp: ellisomat(E)
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
19.
Its isogeny class 144400ba
consists of 2 curves linked by isogenies of
degree 19.
Twists
The minimal quadratic twist of this elliptic curve is 361a1, its twist by $-20$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.76.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.109744.2 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.16462294679520000.1 | \(\Z/3\Z\) | not in database |
| $8$ | 8.0.37636704800000.5 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/7\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $16$ | deg 16 | \(\Z/5\Z\) | not in database |
| $18$ | 18.0.147682003746622037852672000000000.1 | \(\Z/19\Z\) | not in database |
| $20$ | 20.0.366554147091158719162550042716160000000000.1 | \(\Z/11\Z\) | 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
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ss | add | ord | ord | ss | ord | add | ord | ss | ss | ss | ss | ord | ord |
| $\lambda$-invariant(s) | - | 6,2 | - | 0 | 0 | 0,0 | 0 | - | 0 | 0,0 | 0,0 | 0,0 | 0,0 | 0 | 0 |
| $\mu$-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.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.