Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-419372737x-3305723454817\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-419372737xz^2-3305723454817z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-33969191724x-2409770490986448\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-11831, 0)$ | $0$ | $2$ |
Integral points
\( \left(-11831, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 485184 \) | = | $2^{6} \cdot 3 \cdot 7 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $5467770036955120140288$ | = | $2^{19} \cdot 3^{5} \cdot 7 \cdot 19^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{661397832743623417}{443352042} \) | = | $2^{-1} \cdot 3^{-5} \cdot 7^{-1} \cdot 13^{3} \cdot 19^{-4} \cdot 67021^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4872420227904003050195213622$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.97530176236726211088915946407$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0038346664696574$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.436516491234321$ | |||
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: | $\Omega$ | ≈ | $0.033361803769491017987858219207$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 40 $ = $ 2\cdot5\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.3378886031185628780573150732 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
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BSD formula
$$\begin{aligned} 5.337888603 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.033362 \cdot 1.000000 \cdot 40}{2^2} \\ & \approx 5.337888603\end{aligned}$$
Modular invariants
Modular form 485184.2.a.ir
For more coefficients, see the Downloads section to the right.
| Modular degree: | 88473600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 | 8.12.0.12 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3185 & 8 \\ 3184 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2183 & 3184 \\ 2348 & 3159 \end{array}\right),\left(\begin{array}{rr} 460 & 1 \\ 1847 & 6 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3186 & 3187 \end{array}\right),\left(\begin{array}{rr} 2003 & 1996 \\ 2042 & 405 \end{array}\right),\left(\begin{array}{rr} 1072 & 3 \\ 1069 & 2 \end{array}\right),\left(\begin{array}{rr} 1992 & 391 \\ 1205 & 1218 \end{array}\right)$.
The torsion field $K:=\Q(E[3192])$ is a degree-$381250437120$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3192\Z)$.
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 | $4$ | \( 7581 = 3 \cdot 7 \cdot 19^{2} \) |
| $3$ | split multiplicative | $4$ | \( 161728 = 2^{6} \cdot 7 \cdot 19^{2} \) |
| $5$ | good | $2$ | \( 161728 = 2^{6} \cdot 7 \cdot 19^{2} \) |
| $7$ | split multiplicative | $8$ | \( 69312 = 2^{6} \cdot 3 \cdot 19^{2} \) |
| $19$ | additive | $200$ | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 485184ir
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 798d3, its twist by $-152$.
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 $0$.