Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-552x+4992\)
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(homogenize, simplify) |
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\(y^2z=x^3-552xz^2+4992z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-552x+4992\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(16, 16)$ | $0.91242325900191993279637821662$ | $\infty$ |
Integral points
\((16,\pm 16)\)
Invariants
| Conductor: | $N$ | = | \( 6912 \) | = | $2^{8} \cdot 3^{3}$ |
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| Discriminant: | $\Delta$ | = | $-884736$ | = | $-1 \cdot 2^{15} \cdot 3^{3} $ |
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| j-invariant: | $j$ | = | \( -21024576 \) | = | $-1 \cdot 2^{6} \cdot 3^{3} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.21474289625055082682406314224$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.92634415161640823279628831881$ |
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| $abc$ quality: | $Q$ | ≈ | $1.255513869609607$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.4559755571795154$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.91242325900191993279637821662$ |
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| Real period: | $\Omega$ | ≈ | $2.6431580668062179862590065416$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.8233577947450876446906657400 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.823357795 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.643158 \cdot 0.912423 \cdot 2}{1^2} \\ & \approx 4.823357795\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1536 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 8 | 15 | 0 |
| $3$ | $1$ | $II$ | additive | 1 | 3 | 3 | 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 |
|---|---|---|
| $2$ | 2G | 4.4.0.1 |
| $3$ | 3Nn | 9.9.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 72 = 2^{3} \cdot 3^{2} \), index $144$, genus $11$, and generators
$\left(\begin{array}{rr} 53 & 43 \\ 43 & 40 \end{array}\right),\left(\begin{array}{rr} 20 & 29 \\ 67 & 32 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 44 & 63 \\ 45 & 8 \end{array}\right),\left(\begin{array}{rr} 21 & 20 \\ 40 & 69 \end{array}\right),\left(\begin{array}{rr} 30 & 61 \\ 5 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 37 & 36 \\ 36 & 37 \end{array}\right),\left(\begin{array}{rr} 21 & 8 \\ 28 & 69 \end{array}\right)$.
The torsion field $K:=\Q(E[72])$ is a degree-$41472$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/72\Z)$.
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 | $4$ | \( 27 = 3^{3} \) |
| $3$ | additive | $6$ | \( 256 = 2^{8} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 6912a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 6912b1, its twist by $12$.
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 |
|---|---|---|---|
| $3$ | 3.1.216.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.1119744.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.36691771392.3 | \(\Z/3\Z\) | not in database |
| $12$ | 12.2.493024690386763776.33 | \(\Z/4\Z\) | not in database |
| $16$ | 16.0.1346286087882789617664.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | ord | ord | ord | ord | ord | ss | ss | ord | ord | ss | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | 1 | 7 | 1 | 1 | 1 | 1,1 | 1,1 | 1 | 1 | 1,1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | - | 0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.