Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+11803x-572899\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+11803xz^2-572899z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+188853x-36476666\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 428490 \) | = | $2 \cdot 3^{4} \cdot 5 \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-245573775544320$ | = | $-1 \cdot 2^{12} \cdot 3^{4} \cdot 5 \cdot 23^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{15166431}{20480} \) | = | $2^{-12} \cdot 3^{2} \cdot 5^{-1} \cdot 7^{3} \cdot 17^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4469812785520604670557291482$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.48696992563521760881272901335$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9801086025563736$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.0868041345263633$ | |||
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: | $\Omega$ | ≈ | $0.29570780134619915570952464753$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ ( 2^{2} \cdot 3 )\cdot1\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.5484936161543898685142957704 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 3.548493616 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.295708 \cdot 1.000000 \cdot 12}{1^2} \\ & \approx 3.548493616\end{aligned}$$
Modular invariants
Modular form 428490.2.a.bw
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1568160 |
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| $ \Gamma_0(N) $-optimal: | not computed* (one of 2 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $23$ | $1$ | $I_0^{*}$ | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1375 & 6 \\ 1374 & 7 \end{array}\right),\left(\begin{array}{rr} 1266 & 1081 \\ 805 & 231 \end{array}\right),\left(\begin{array}{rr} 691 & 966 \\ 1173 & 139 \end{array}\right),\left(\begin{array}{rr} 277 & 966 \\ 1311 & 139 \end{array}\right),\left(\begin{array}{rr} 179 & 0 \\ 0 & 1379 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1380])$ is a degree-$36933304320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1380\Z)$.
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 | $4$ | \( 214245 = 3^{4} \cdot 5 \cdot 23^{2} \) |
| $3$ | additive | $8$ | \( 2645 = 5 \cdot 23^{2} \) |
| $5$ | split multiplicative | $6$ | \( 85698 = 2 \cdot 3^{4} \cdot 23^{2} \) |
| $23$ | additive | $266$ | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 428490.bw
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 810.b2, its twist by $69$.
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 $0$.