Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-984x-11864\)
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(homogenize, simplify) |
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\(y^2z=x^3-984xz^2-11864z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-984x-11864\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-18, 4)$ | $1.3544921056018688036569182365$ | $\infty$ |
| $(-158/9, 8/27)$ | $3.5476071789719024524778040687$ | $\infty$ |
Integral points
\((-18,\pm 4)\), \((105,\pm 1021)\), \((129,\pm 1417)\)
Invariants
| Conductor: | $N$ | = | \( 131904 \) | = | $2^{6} \cdot 3^{2} \cdot 229$ |
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| Discriminant: | $\Delta$ | = | $170947584$ | = | $2^{10} \cdot 3^{6} \cdot 229 $ |
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| j-invariant: | $j$ | = | \( \frac{141150208}{229} \) | = | $2^{11} \cdot 41^{3} \cdot 229^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.47664699691031924222347297315$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.65028179789035669465517641319$ |
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| $abc$ quality: | $Q$ | ≈ | $0.800753878141722$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $2.738672367912796$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8032232539521565736283491882$ |
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| Real period: | $\Omega$ | ≈ | $0.85249747066623934505554123999$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $8.1894713500789545898613336711 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.189471350 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.852497 \cdot 4.803223 \cdot 2}{1^2} \\ & \approx 8.189471350\end{aligned}$$
Modular invariants
Modular form 131904.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 84480 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | 1 | 6 | 10 | 0 |
| $3$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $229$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 458 = 2 \cdot 229 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 457 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 235 & 2 \\ 235 & 3 \end{array}\right),\left(\begin{array}{rr} 457 & 2 \\ 456 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[458])$ is a degree-$8213991840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/458\Z)$.
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 | $2$ | \( 2061 = 3^{2} \cdot 229 \) |
| $3$ | additive | $2$ | \( 14656 = 2^{6} \cdot 229 \) |
| $229$ | nonsplit multiplicative | $230$ | \( 576 = 2^{6} \cdot 3^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 131904bb consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 1832b1, its twist by $-24$.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.229.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.6.12008989.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 229 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord | nonsplit |
| $\lambda$-invariant(s) | - | - | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.