Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-31059x+497448\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-31059xz^2+497448z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-496947x+31339726\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(168, -84)$ | $0$ | $2$ |
Integral points
\( \left(168, -84\right) \)
Invariants
| Conductor: | $N$ | = | \( 495495 \) | = | $3^{2} \cdot 5 \cdot 7 \cdot 11^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $1813976537237865$ | = | $3^{8} \cdot 5 \cdot 7^{4} \cdot 11^{6} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{2565726409}{1404585} \) | = | $3^{-2} \cdot 5^{-1} \cdot 7^{-4} \cdot 13^{-1} \cdot 37^{6}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6185868854649287614041234510$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.12966689526831135632447095644$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0367988511083188$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2520046104641795$ | |||
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: | $\Omega$ | ≈ | $0.40890175516301489275174692599$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.6356070206520595710069877040 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.635607021 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.408902 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 1.635607021\end{aligned}$$
Modular invariants
Modular form 495495.2.a.dy
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2621440 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $13$ | $1$ | $I_{1}$ | 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120120 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 43679 & 0 \\ 0 & 120119 \end{array}\right),\left(\begin{array}{rr} 33496 & 10923 \\ 58245 & 83722 \end{array}\right),\left(\begin{array}{rr} 51481 & 69168 \\ 60324 & 36433 \end{array}\right),\left(\begin{array}{rr} 90553 & 116028 \\ 43230 & 8647 \end{array}\right),\left(\begin{array}{rr} 40039 & 0 \\ 0 & 120119 \end{array}\right),\left(\begin{array}{rr} 38644 & 83721 \\ 12903 & 58246 \end{array}\right),\left(\begin{array}{rr} 120113 & 8 \\ 120112 & 9 \end{array}\right),\left(\begin{array}{rr} 25939 & 25938 \\ 52338 & 17755 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 120114 & 120115 \end{array}\right)$.
The torsion field $K:=\Q(E[120120])$ is a degree-$514198484287488000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120120\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 70785 = 3^{2} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $3$ | additive | $8$ | \( 55055 = 5 \cdot 7 \cdot 11^{2} \cdot 13 \) |
| $5$ | split multiplicative | $6$ | \( 99099 = 3^{2} \cdot 7 \cdot 11^{2} \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 70785 = 3^{2} \cdot 5 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $62$ | \( 4095 = 3^{2} \cdot 5 \cdot 7 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 38115 = 3^{2} \cdot 5 \cdot 7 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 495495dy
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 1365c1, its twist by $33$.
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 $0$.