Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-96x+386\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-96xz^2+386z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1539x+23166\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5, 1)$ | $1.3643323439534196824257105454$ | $\infty$ |
| $(7, 1)$ | $0$ | $3$ |
Integral points
\( \left(-11, 10\right) \), \( \left(-11, 1\right) \), \( \left(5, 1\right) \), \( \left(5, -6\right) \), \( \left(7, 1\right) \), \( \left(7, -8\right) \), \( \left(1015, 31816\right) \), \( \left(1015, -32831\right) \)
Invariants
| Conductor: | $N$ | = | \( 486 \) | = | $2 \cdot 3^{5}$ |
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| Discriminant: | $\Delta$ | = | $354294$ | = | $2 \cdot 3^{11} $ |
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| j-invariant: | $j$ | = | \( \frac{555579}{2} \) | = | $2^{-1} \cdot 3^{4} \cdot 19^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.073919984185448757563656826360$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0809812487978826413426316269$ |
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| $abc$ quality: | $Q$ | ≈ | $1.024011249937178$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.091763276415509$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.3643323439534196824257105454$ |
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| Real period: | $\Omega$ | ≈ | $3.0412502684676547559643301877$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.3830920357758140993531487215 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.383092036 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 3.041250 \cdot 1.364332 \cdot 3}{3^2} \\ & \approx 1.383092036\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 108 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $3$ | $IV^{*}$ | additive | 1 | 5 | 11 | 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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 24.16.0-24.c.1.8, level \( 24 = 2^{3} \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22 & 21 \\ 23 & 14 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 6 \\ 15 & 19 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 21 & 19 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 19 & 6 \\ 18 & 7 \end{array}\right)$.
The torsion field $K:=\Q(E[24])$ is a degree-$4608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 243 = 3^{5} \) |
| $3$ | additive | $8$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 486b
consists of 2 curves linked by isogenies of
degree 3.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.1944.1 | \(\Z/6\Z\) | 3.3.1944.1-36.1-d4 |
| $6$ | 6.6.90699264.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.2834352.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.167365651248.3 | \(\Z/9\Z\) | not in database |
| $12$ | 12.4.12635683568857645056.5 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.373062239528737541455872.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | nonsplit | add | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | - | 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 |
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.