Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+49236675x-577064080375\)
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(homogenize, simplify) |
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\(y^2z=x^3+49236675xz^2-577064080375z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+49236675x-577064080375\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 476100 \) | = | $2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}$ |
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| Discriminant: | $\Delta$ | = | $-151496444754821777343750000$ | = | $-1 \cdot 2^{4} \cdot 3^{6} \cdot 5^{18} \cdot 23^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{489277573376}{5615234375} \) | = | $2^{8} \cdot 5^{-12} \cdot 17^{3} \cdot 23^{-1} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7044977086173015617397267113$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.55167643991497324686593730317$ |
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| $abc$ quality: | $Q$ | ≈ | $1.147813072895248$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.177357906315339$ | |||
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: | $\Omega$ | ≈ | $0.028426251771430016962724871772$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot1\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0233450637714806106580953838 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.023345064 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.028426 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.023345064\end{aligned}$$
Modular invariants
Modular form 476100.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 164229120 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{12}^{*}$ | additive | 1 | 2 | 18 | 12 |
| $23$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 |
|---|---|---|
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 664 & 435 \\ 335 & 269 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 179 & 270 \\ 675 & 119 \end{array}\right),\left(\begin{array}{rr} 685 & 6 \\ 684 & 7 \end{array}\right),\left(\begin{array}{rr} 413 & 0 \\ 0 & 689 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[690])$ is a degree-$2308331520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/690\Z)$.
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 | $2$ | \( 119025 = 3^{2} \cdot 5^{2} \cdot 23^{2} \) |
| $3$ | additive | $2$ | \( 52900 = 2^{2} \cdot 5^{2} \cdot 23^{2} \) |
| $5$ | additive | $18$ | \( 19044 = 2^{2} \cdot 3^{2} \cdot 23^{2} \) |
| $23$ | additive | $288$ | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 476100.c
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 460.c2, its twist by $345$.
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 $0$.