Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-594017004x+5572444750544\)
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(homogenize, simplify) |
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\(y^2z=x^3-594017004xz^2+5572444750544z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-594017004x+5572444750544\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(14242, 34992)$ | $1.6470905363313071925009299466$ | $\infty$ |
| $(14050, 3312)$ | $4.2120066383811445879539647957$ | $\infty$ |
Integral points
\((-1067,\pm 2490993)\), \((14050,\pm 3312)\), \((14242,\pm 34992)\), \((14365,\pm 60453)\)
Invariants
| Conductor: | $N$ | = | \( 479808 \) | = | $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $40641579901815028383744$ | = | $2^{16} \cdot 3^{17} \cdot 7^{10} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{1717641340122148}{3011499} \) | = | $2^{2} \cdot 3^{-11} \cdot 7^{2} \cdot 17^{-1} \cdot 53^{3} \cdot 389^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5975719081905252751357271710$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.50247773223044892929400110439$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0102533511357885$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.520990628429188$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.9343418105517436377082347436$ |
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| Real period: | $\Omega$ | ≈ | $0.098129334666974775743133431304$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2^{2}\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.887397571565245595616470781 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.887397572 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.098129 \cdot 6.934342 \cdot 16}{1^2} \\ & \approx 10.887397572\end{aligned}$$
Modular invariants
Modular form 479808.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 113541120 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{6}^{*}$ | additive | -1 | 6 | 16 | 0 |
| $3$ | $4$ | $I_{11}^{*}$ | additive | -1 | 2 | 17 | 11 |
| $7$ | $1$ | $II^{*}$ | additive | -1 | 2 | 10 | 0 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 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 \( 204 = 2^{2} \cdot 3 \cdot 17 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 203 & 0 \end{array}\right),\left(\begin{array}{rr} 103 & 2 \\ 103 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 203 & 2 \\ 202 & 3 \end{array}\right),\left(\begin{array}{rr} 37 & 2 \\ 37 & 3 \end{array}\right),\left(\begin{array}{rr} 137 & 2 \\ 137 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[204])$ is a degree-$180486144$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/204\Z)$.
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 | $2$ | \( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \) |
| $3$ | additive | $8$ | \( 53312 = 2^{6} \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $20$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 479808u consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 19992n1, its twist by $-168$.
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.