Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-470864x-37451116\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-470864xz^2-37451116z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-38140011x-27187443558\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 456304 \) | = | $2^{4} \cdot 19^{2} \cdot 79$ |
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| Minimal Discriminant: | $\Delta$ | = | $6080549125451677696$ | = | $2^{18} \cdot 19^{6} \cdot 79^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{59914169497}{31554496} \) | = | $2^{-6} \cdot 7^{3} \cdot 13^{3} \cdot 43^{3} \cdot 79^{-3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2959185951908960081973435313$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.13055192504773046877559769390$ |
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| $abc$ quality: | $Q$ | ≈ | $0.967983438848399$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.898466126805232$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.19331644990668038916620944337$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3197973988801646699945133205 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.319797399 \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 0.193316 \cdot 1.000000 \cdot 12}{1^2} \\ & \approx 2.319797399\end{aligned}$$
Modular invariants
Modular form 456304.2.a.bd
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6842880 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}^{*}$ | additive | -1 | 4 | 18 | 6 |
| $19$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $79$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 |
|---|---|---|---|
| $3$ | 3Cs | 3.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 54036 = 2^{2} \cdot 3^{2} \cdot 19 \cdot 79 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20236 & 31293 \\ 41211 & 17044 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 49781 & 8550 \\ 31122 & 2737 \end{array}\right),\left(\begin{array}{rr} 22742 & 22743 \\ 4275 & 31292 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 54019 & 18 \\ 54018 & 19 \end{array}\right),\left(\begin{array}{rr} 45503 & 0 \\ 0 & 54035 \end{array}\right)$.
The torsion field $K:=\Q(E[54036])$ is a degree-$12270715519795200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/54036\Z)$.
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 | $2$ | \( 28519 = 19^{2} \cdot 79 \) |
| $3$ | good | $2$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 1264 = 2^{4} \cdot 79 \) |
| $79$ | split multiplicative | $80$ | \( 5776 = 2^{4} \cdot 19^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 456304bd
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 158d1, its twist by $76$.
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 $0$.