Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-16702875x+24923666250\)
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(homogenize, simplify) |
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\(y^2z=x^3-16702875xz^2+24923666250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-16702875x+24923666250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3325, 78400)$ | $0.90277851340182860653199446338$ | $\infty$ |
| $(-4025, 164150)$ | $3.5276156315770214603067054397$ | $\infty$ |
Integral points
\((-4025,\pm 164150)\), \((1414,\pm 64288)\), \((1789,\pm 27712)\), \((3325,\pm 78400)\), \((4375,\pm 188650)\)
Invariants
| Conductor: | $N$ | = | \( 529200 \) | = | $2^{4} \cdot 3^{3} \cdot 5^{2} \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $29877897588940800000000$ | = | $2^{19} \cdot 3^{11} \cdot 5^{8} \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{15454515}{896} \) | = | $2^{-7} \cdot 3 \cdot 5 \cdot 7^{-1} \cdot 101^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0664091451213044322362905135$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.67971298286813166324643233568$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8762791182580077$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.666993646103279$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0231273075091674409273844675$ |
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| Real period: | $\Omega$ | ≈ | $0.11583495938214464143220116365$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot1\cdot3\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $16.808823785874081771645361818 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 16.808823786 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.115835 \cdot 3.023127 \cdot 48}{1^2} \\ & \approx 16.808823786\end{aligned}$$
Modular invariants
Modular form 529200.2.a.dg
For more coefficients, see the Downloads section to the right.
| Modular degree: | 34836480 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{11}^{*}$ | additive | -1 | 4 | 19 | 7 |
| $3$ | $1$ | $II^{*}$ | additive | 1 | 3 | 11 | 0 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 168 = 2^{3} \cdot 3 \cdot 7 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 167 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 113 & 2 \\ 113 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 85 & 2 \\ 85 & 3 \end{array}\right),\left(\begin{array}{rr} 167 & 2 \\ 166 & 3 \end{array}\right),\left(\begin{array}{rr} 127 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 73 & 2 \\ 73 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[168])$ is a degree-$74317824$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/168\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 33075 = 3^{3} \cdot 5^{2} \cdot 7^{2} \) |
| $3$ | additive | $4$ | \( 19600 = 2^{4} \cdot 5^{2} \cdot 7^{2} \) |
| $5$ | additive | $14$ | \( 21168 = 2^{4} \cdot 3^{3} \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 10800 = 2^{4} \cdot 3^{3} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 529200.dg consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 9450.x1, its twist by $140$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.