Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-319932136524x+69652211962529392\)
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(homogenize, simplify) |
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\(y^2z=x^3-319932136524xz^2+69652211962529392z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-319932136524x+69652211962529392\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(326564, 0)$ | $0$ | $2$ |
Integral points
\( \left(326564, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 479808 \) | = | $2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $47785514287848357888$ | = | $2^{17} \cdot 3^{12} \cdot 7^{9} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{322159999717985454060440834}{4250799} \) | = | $2 \cdot 3^{-6} \cdot 7^{-3} \cdot 11^{6} \cdot 13^{3} \cdot 17^{-1} \cdot 191^{3} \cdot 1811^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.7598421146487774324276084656$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2556223899938100791695639700$ |
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| $abc$ quality: | $Q$ | ≈ | $1.108049552899778$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.963285934649332$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.046246468338023587578628871585$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.3297457203376983056612787541 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.329745720 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.046246 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 3.329745720\end{aligned}$$
Modular invariants
Modular form 479808.2.a.oj
For more coefficients, see the Downloads section to the right.
| Modular degree: | 849346560 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 3 curves in this isogeny class which might be optimal) | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{7}^{*}$ | additive | 1 | 6 | 17 | 0 |
| $3$ | $4$ | $I_{6}^{*}$ | additive | -1 | 2 | 12 | 6 |
| $7$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 |
|---|---|---|
| $2$ | 2B | 8.12.0.9 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2856 = 2^{3} \cdot 3 \cdot 7 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 568 & 3 \\ 117 & 1906 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1903 & 0 \\ 0 & 2855 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 2850 & 2851 \end{array}\right),\left(\begin{array}{rr} 536 & 2853 \\ 2715 & 950 \end{array}\right),\left(\begin{array}{rr} 2849 & 8 \\ 2848 & 9 \end{array}\right),\left(\begin{array}{rr} 1072 & 603 \\ 123 & 2056 \end{array}\right),\left(\begin{array}{rr} 1544 & 111 \\ 1317 & 2282 \end{array}\right)$.
The torsion field $K:=\Q(E[2856])$ is a degree-$242573377536$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2856\Z)$.
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 | $4$ | \( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \) |
| $3$ | additive | $6$ | \( 53312 = 2^{6} \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $32$ | \( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 479808.oj
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2856.c1, its twist by $168$.
Iwasawa invariants
No Iwasawa invariant data is available for this curve.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.