Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-160563x-5872312\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-160563xz^2-5872312z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-208089027x-273354309954\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1671/4, -1675/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 481390 \) | = | $2 \cdot 5 \cdot 7 \cdot 13 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $250089336471870350$ | = | $2 \cdot 5^{2} \cdot 7 \cdot 13^{6} \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3092354182009}{1689383150} \) | = | $2^{-1} \cdot 5^{-2} \cdot 7^{-1} \cdot 13^{-6} \cdot 17^{3} \cdot 857^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0291589096817256695997576732$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.46141180171715082419638125729$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9448924965191774$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.635841076910515$ | |||
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: | $\Omega$ | ≈ | $0.25467956812845704653374383869$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 1\cdot2\cdot1\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.5280774087707422792024630322 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.528077409 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.254680 \cdot 1.000000 \cdot 24}{2^2} \\ & \approx 1.528077409\end{aligned}$$
Modular invariants
Modular form 481390.2.a.h
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6842880 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 4 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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $23$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 251160 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 23 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 120130 & 207483 \\ 16353 & 131032 \end{array}\right),\left(\begin{array}{rr} 135241 & 76452 \\ 221766 & 207553 \end{array}\right),\left(\begin{array}{rr} 150697 & 76452 \\ 63342 & 207553 \end{array}\right),\left(\begin{array}{rr} 50969 & 54602 \\ 34638 & 76453 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 251110 & 251151 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 251149 & 12 \\ 251148 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 12490 & 207483 \\ 106053 & 131032 \end{array}\right),\left(\begin{array}{rr} 177906 & 212497 \\ 52325 & 10466 \end{array}\right),\left(\begin{array}{rr} 109199 & 0 \\ 0 & 251159 \end{array}\right)$.
The torsion field $K:=\Q(E[251160])$ is a degree-$5203688660989378560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/251160\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 3703 = 7 \cdot 23^{2} \) |
| $3$ | good | $2$ | \( 37030 = 2 \cdot 5 \cdot 7 \cdot 23^{2} \) |
| $5$ | split multiplicative | $6$ | \( 96278 = 2 \cdot 7 \cdot 13 \cdot 23^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 68770 = 2 \cdot 5 \cdot 13 \cdot 23^{2} \) |
| $13$ | split multiplicative | $14$ | \( 37030 = 2 \cdot 5 \cdot 7 \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 910 = 2 \cdot 5 \cdot 7 \cdot 13 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 481390h
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 910c2, its twist by $-23$.
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 $0$.