Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-131651088x+555980666292\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-131651088xz^2+555980666292z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-170619810075x+25940345825949750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5448, 18570)$ | $3.8923482543498280308522873037$ | $\infty$ |
| $(-52737/4, 52737/8)$ | $0$ | $2$ |
Integral points
\( \left(5448, 18570\right) \), \( \left(5448, -24018\right) \)
Invariants
| Conductor: | $N$ | = | \( 481650 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 13^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $12491047040168819179687500$ | = | $2^{2} \cdot 3^{20} \cdot 5^{10} \cdot 13^{6} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( \frac{3345930611358906241}{165622259047500} \) | = | $2^{-2} \cdot 3^{-20} \cdot 5^{-4} \cdot 11^{6} \cdot 19^{-1} \cdot 47^{3} \cdot 263^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5746820228682300787312556662$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4874883879204115234041322788$ |
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| $abc$ quality: | $Q$ | ≈ | $1.081271340032757$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.173919748932871$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.8923482543498280308522873037$ |
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| Real period: | $\Omega$ | ≈ | $0.070267567455686633420665227829$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 320 $ = $ 2\cdot( 2^{2} \cdot 5 )\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $21.880467481884052377749397594 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 21.880467482 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.070268 \cdot 3.892348 \cdot 320}{2^2} \\ & \approx 21.880467482\end{aligned}$$
Modular invariants
Modular form 481650.2.a.is
For more coefficients, see the Downloads section to the right.
| Modular degree: | 176947200 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 (conditional*) |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $20$ | $I_{20}$ | split multiplicative | -1 | 1 | 20 | 20 |
| $5$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{1}$ | nonsplit 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 \( 29640 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 859 & 858 \\ 18538 & 17395 \end{array}\right),\left(\begin{array}{rr} 29633 & 8 \\ 29632 & 9 \end{array}\right),\left(\begin{array}{rr} 26768 & 4563 \\ 16445 & 11402 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 29634 & 29635 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 9881 & 2288 \\ 25844 & 9153 \end{array}\right),\left(\begin{array}{rr} 11688 & 5993 \\ 11713 & 2640 \end{array}\right),\left(\begin{array}{rr} 11399 & 0 \\ 0 & 29639 \end{array}\right),\left(\begin{array}{rr} 17783 & 27352 \\ 25532 & 20487 \end{array}\right)$.
The torsion field $K:=\Q(E[29640])$ is a degree-$2379002727628800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/29640\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 80275 = 5^{2} \cdot 13^{2} \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 160550 = 2 \cdot 5^{2} \cdot 13^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 6422 = 2 \cdot 13^{2} \cdot 19 \) |
| $13$ | additive | $86$ | \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 25350 = 2 \cdot 3 \cdot 5^{2} \cdot 13^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 481650is
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 570i3, its twist by $65$.
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.