Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-449287x+220349329\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-449287xz^2+220349329z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-582276627x+10289352439854\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(28, 14401\right) \) | $0.35113732239514224078607204492$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([28:14401:1]\) | $0.35113732239514224078607204492$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1023, 3113640\right) \) | $0.35113732239514224078607204492$ | $\infty$ |
Integral points
\( \left(-157, 17021\right) \), \( \left(-157, -16864\right) \), \( \left(28, 14401\right) \), \( \left(28, -14429\right) \), \( \left(183, 11921\right) \), \( \left(183, -12104\right) \), \( \left(28858, 4886671\right) \), \( \left(28858, -4915529\right) \)
\([-157:17021:1]\), \([-157:-16864:1]\), \([28:14401:1]\), \([28:-14429:1]\), \([183:11921:1]\), \([183:-12104:1]\), \([28858:4886671:1]\), \([28858:-4915529:1]\)
\((-5637,\pm 3659580)\), \((1023,\pm 3113640)\), \((6603,\pm 2594700)\), \((1038903,\pm 1058637600)\)
Invariants
| Conductor: | $N$ | = | \( 490110 \) | = | $2 \cdot 3 \cdot 5 \cdot 17 \cdot 31^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-15206565726826087500$ | = | $-1 \cdot 2^{2} \cdot 3^{2} \cdot 5^{5} \cdot 17^{3} \cdot 31^{7} $ |
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| j-invariant: | $j$ | = | \( -\frac{11301253512121}{17134087500} \) | = | $-1 \cdot 2^{-2} \cdot 3^{-2} \cdot 5^{-5} \cdot 17^{-3} \cdot 31^{-1} \cdot 22441^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3722985652031017888143805422$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.65530496296052866584979837993$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8864236695374322$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9646440581952866$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.35113732239514224078607204492$ |
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| Real period: | $\Omega$ | ≈ | $0.19883380645943936616541453737$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ 2\cdot2\cdot5\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.5854376321441181188086485443 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.585437632 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.198834 \cdot 0.351137 \cdot 80}{1^2} \\ & \approx 5.585437632\end{aligned}$$
Modular invariants
Modular form 490110.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 13824000 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $17$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $31$ | $4$ | $I_{1}^{*}$ | 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 \( 5270 = 2 \cdot 5 \cdot 17 \cdot 31 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 5269 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2791 & 2 \\ 2791 & 3 \end{array}\right),\left(\begin{array}{rr} 4217 & 2 \\ 4217 & 3 \end{array}\right),\left(\begin{array}{rr} 5269 & 2 \\ 5268 & 3 \end{array}\right),\left(\begin{array}{rr} 4591 & 2 \\ 4591 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[5270])$ is a degree-$100711268352000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5270\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 81685 = 5 \cdot 17 \cdot 31^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 9610 = 2 \cdot 5 \cdot 31^{2} \) |
| $5$ | split multiplicative | $6$ | \( 98022 = 2 \cdot 3 \cdot 17 \cdot 31^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 28830 = 2 \cdot 3 \cdot 5 \cdot 31^{2} \) |
| $31$ | additive | $512$ | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 490110q consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 15810h1, its twist by $-31$.
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.