Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-364369x-179858757\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-364369xz^2-179858757z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-29513916x-131028492132\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(11637694/7569, 35443801997/658503)$ | $10.250204896075245615849371877$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 479408 \) | = | $2^{4} \cdot 19^{2} \cdot 83$ |
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| Discriminant: | $\Delta$ | = | $-10859942892106296064$ | = | $-1 \cdot 2^{8} \cdot 19^{7} \cdot 83^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{444209247232}{901708099} \) | = | $-1 \cdot 2^{10} \cdot 19^{-1} \cdot 83^{-4} \cdot 757^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3419283954256860741473431159$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.40761078546916897119800798565$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8671875144785367$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.940057232966936$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.250204896075245615849371877$ |
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| Real period: | $\Omega$ | ≈ | $0.091241912587383425723333128447$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.4819863930437373766597903708 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.481986393 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.091242 \cdot 10.250205 \cdot 8}{1^2} \\ & \approx 7.481986393\end{aligned}$$
Modular invariants
Modular form 479408.2.a.i
For more coefficients, see the Downloads section to the right.
| Modular degree: | 11335680 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | 1 | 4 | 8 | 0 |
| $19$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
| $83$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 label 38.2.0.a.1, level \( 38 = 2 \cdot 19 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 37 & 2 \\ 36 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 21 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 1 \\ 37 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[38])$ is a degree-$369360$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/38\Z)$.
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 | $2$ | \( 361 = 19^{2} \) |
| $19$ | additive | $200$ | \( 1328 = 2^{4} \cdot 83 \) |
| $83$ | split multiplicative | $84$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 479408i consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 12616b1, its twist by $76$.
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.