Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-50226x-413820\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-50226xz^2-413820z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-65092923x-19111907178\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(396, 6270\right)\) |
$\hat{h}(P)$ | ≈ | $0.24741524217734823180984852438$ |
Torsion generators
\( \left(-220, 110\right) \), \( \left(228, -114\right) \)
Integral points
\( \left(-220, 110\right) \), \( \left(-192, 1566\right) \), \( \left(-192, -1374\right) \), \( \left(-132, 2046\right) \), \( \left(-132, -1914\right) \), \( \left(-66, 1650\right) \), \( \left(-66, -1584\right) \), \( \left(-24, 894\right) \), \( \left(-24, -870\right) \), \( \left(-22, 836\right) \), \( \left(-22, -814\right) \), \( \left(228, -114\right) \), \( \left(264, 2046\right) \), \( \left(264, -2310\right) \), \( \left(396, 6270\right) \), \( \left(396, -6666\right) \), \( \left(804, 21486\right) \), \( \left(804, -22290\right) \), \( \left(858, 23826\right) \), \( \left(858, -24684\right) \), \( \left(3168, 176286\right) \), \( \left(3168, -179454\right) \), \( \left(3630, 216480\right) \), \( \left(3630, -220110\right) \), \( \left(728508, 621436926\right) \), \( \left(728508, -622165434\right) \)
Invariants
Conductor: | \( 16170 \) | = | $2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $8036491376390400 $ | = | $2^{8} \cdot 3^{6} \cdot 5^{2} \cdot 7^{6} \cdot 11^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{119102750067601}{68309049600} \) | = | $2^{-8} \cdot 3^{-6} \cdot 5^{-2} \cdot 11^{-4} \cdot 49201^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $1.7415675175406415939340007231\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.76861244301298494138132435138\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0644140254035126\dots$ | |||
Szpiro ratio: | $4.549258546901542\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.24741524217734823180984852438\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.34613549424759314764540256556\dots$ | 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: | $ 1536 $ = $ 2^{3}\cdot( 2 \cdot 3 )\cdot2\cdot2^{2}\cdot2^{2} $ | 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: | $4$ | 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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Analytic order of Ш: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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Special value: | $ L'(E,1) $ ≈ $ 8.2213629250026609037120940103 $ | 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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BSD formula
$\displaystyle 8.221362925 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.346135 \cdot 0.247415 \cdot 1536}{4^2} \approx 8.221362925$
Modular invariants
Modular form 16170.2.a.bz
For more coefficients, see the Downloads section to the right.
Modular degree: | 147456 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $8$ | $I_{8}$ | Split multiplicative | -1 | 1 | 8 | 8 |
$3$ | $6$ | $I_{6}$ | Split multiplicative | -1 | 1 | 6 | 6 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$11$ | $4$ | $I_{4}$ | Split multiplicative | -1 | 1 | 4 | 4 |
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 |
---|---|---|
$2$ | 2Cs | 4.12.0.3 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 799 & 3962 \\ 8694 & 6595 \end{array}\right),\left(\begin{array}{rr} 9233 & 8 \\ 9232 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 9236 & 9237 \end{array}\right),\left(\begin{array}{rr} 8807 & 2646 \\ 6594 & 6595 \end{array}\right),\left(\begin{array}{rr} 3305 & 336 \\ 4284 & 7253 \end{array}\right),\left(\begin{array}{rr} 3963 & 994 \\ 1316 & 4285 \end{array}\right),\left(\begin{array}{rr} 3959 & 0 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 2521 & 6608 \\ 8764 & 7953 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$4904976384000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 16170.bz
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 330.d5, its twist by $-7$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{7}, \sqrt{-15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{7}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.31116960000.7 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.21970650625.1 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.728933458176.4 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | split | split | nonsplit | add | split | ord | ord | ord | ss | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 5 | 2 | 1 | - | 2 | 1 | 1 | 1 | 1,1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
$\mu$-invariant(s) | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.