Minimal Weierstrass equation
sage: E = EllipticCurve([1, 0, 0, -498053228, -6167895603312])
gp: E = ellinit([1, 0, 0, -498053228, -6167895603312])
magma: E := EllipticCurve([1, 0, 0, -498053228, -6167895603312]);
Minimal equation
Minimal equation
Simplified equation
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\(y^2+xy=x^3-498053228x-6167895603312\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-498053228xz^2-6167895603312z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-645476983515x-287767400837174154\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
sage: E.torsion_subgroup().gens()
gp: elltors(E)
magma: TorsionSubgroup(E);
\( \left(\frac{107839}{4}, -\frac{107839}{8}\right) \)
Integral points
sage: E.integral_points()
magma: IntegralPoints(E);
None
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 189618 \) | = | $2 \cdot 3 \cdot 11 \cdot 13^{2} \cdot 17$ |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | $-8527408942700901249012436992 $ | = | $-1 \cdot 2^{12} \cdot 3^{2} \cdot 11^{2} \cdot 13^{18} \cdot 17 $ |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{2830680648734534916567625}{1766676274677722124288} \) | = | $-1 \cdot 2^{-12} \cdot 3^{-2} \cdot 5^{3} \cdot 7^{3} \cdot 11^{-2} \cdot 13^{-12} \cdot 17^{-1} \cdot 4041683^{3}$ |
| Endomorphism ring: | $\Z$ | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | $4.0611140885299172870306205522\dots$ | ||
| Stable Faltings height: | $2.7786394097991489190038768314\dots$ | ||
BSD invariants
sage: E.rank()
magma: Rank(E);
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| Analytic rank: | $0$ | ||
sage: E.regulator()
magma: Regulator(E);
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| Regulator: | $1$ | ||
sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | $0.015539314208347085820905596224\dots$ | ||
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | $ 192 $ = $ ( 2^{2} \cdot 3 )\cdot2\cdot2\cdot2^{2}\cdot1 $ | ||
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | $2$ | ||
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | $9$ = $3^2$ (exact) | ||
sage: r = E.rank();
gp: ar = ellanalyticrank(E);
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Special value: | $ L(E,1) $ ≈ $ 6.7129837380059410746312175688 $ | ||
Modular invariants
Modular form 189618.2.a.br
sage: E.q_eigenform(20)
gp: xy = elltaniyama(E);
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
magma: ModularForm(E);
\( q + q^{2} + q^{3} + q^{4} + q^{6} + 4 q^{7} + q^{8} + q^{9} - q^{11} + q^{12} + 4 q^{14} + q^{16} + q^{17} + q^{18} - 2 q^{19} + O(q^{20}) \)
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For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 153280512 | ||
| $ \Gamma_0(N) $-optimal: | no | ||
| Manin constant: | 1 | ||
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
sage: E.local_data()
gp: ellglobalred(E)[5]
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | Split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
| $11$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{12}^{*}$ | Additive | 1 | 2 | 18 | 12 |
| $17$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
sage: rho = E.galois_representation();
sage: [rho.image_type(p) for p in rho.non_surjective()]
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
$p$-adic regulators
sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
All $p$-adic regulators are identically $1$ since the rank is $0$.
No Iwasawa invariant data is available for this curve.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 189618.br
consists of 4 curves linked by isogenies of
degrees dividing 6.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-39}) \) | \(\Z/6\Z\) | Not in database |
| $2$ | \(\Q(\sqrt{-17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
| $4$ | 4.2.12514788.1 | \(\Z/4\Z\) | Not in database |
| $4$ | \(\Q(\sqrt{-17}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.2.158638659079147533.1 | \(\Z/6\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
| $8$ | Deg 8 | \(\Z/12\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
| $12$ | Deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
| $18$ | 18.0.193643079756895602999529258711396024499161761922904064.1 | \(\Z/18\Z\) | 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.