Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-246380x+42849263\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-246380xz^2+42849263z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3942075x+2738410774\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(211, 367\right) \) | $2.6405146814202452011247150075$ | $\infty$ |
| \( \left(-219, 9397\right) \) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([211:367:1]\) | $2.6405146814202452011247150075$ | $\infty$ |
| \([-219:9397:1]\) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(843, 3784\right) \) | $2.6405146814202452011247150075$ | $\infty$ |
| \( \left(-877, 74304\right) \) | $0$ | $6$ |
Integral points
\( \left(-299, 9621\right) \), \( \left(-299, -9323\right) \), \( \left(-219, 9397\right) \), \( \left(-219, -9179\right) \), \( \left(211, 367\right) \), \( \left(211, -579\right) \), \( \left(213, -107\right) \), \( \left(429, 3781\right) \), \( \left(429, -4211\right) \), \( \left(469, 5269\right) \), \( \left(469, -5739\right) \), \( \left(55509, 13049749\right) \), \( \left(55509, -13105259\right) \)
\([-299:9621:1]\), \([-299:-9323:1]\), \([-219:9397:1]\), \([-219:-9179:1]\), \([211:367:1]\), \([211:-579:1]\), \([213:-107:1]\), \([429:3781:1]\), \([429:-4211:1]\), \([469:5269:1]\), \([469:-5739:1]\), \([55509:13049749:1]\), \([55509:-13105259:1]\)
\((-1197,\pm 75776)\), \((-877,\pm 74304)\), \((843,\pm 3784)\), \( \left(851, 0\right) \), \((1715,\pm 31968)\), \((1875,\pm 44032)\), \((222035,\pm 104620032)\)
Invariants
| Conductor: | $N$ | = | \( 14706 \) | = | $2 \cdot 3^{2} \cdot 19 \cdot 43$ |
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| Minimal Discriminant: | $\Delta$ | = | $166283402079633408$ | = | $2^{24} \cdot 3^{8} \cdot 19 \cdot 43^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{2268876641163765625}{228097945239552} \) | = | $2^{-24} \cdot 3^{-2} \cdot 5^{6} \cdot 19^{-1} \cdot 43^{-3} \cdot 52561^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0405571993305561590616966615$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4912510549965013133640740430$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0668830281012522$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.091437576807084$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.6405146814202452011247150075$ |
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| Real period: | $\Omega$ | ≈ | $0.31322167261812882862849766829$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 288 $ = $ ( 2^{3} \cdot 3 )\cdot2^{2}\cdot1\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.6165314006973982683912510174 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.616531401 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.313222 \cdot 2.640515 \cdot 288}{6^2} \\ & \approx 6.616531401\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 165888 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $24$ | $I_{24}$ | split multiplicative | -1 | 1 | 24 | 24 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $43$ | $3$ | $I_{3}$ | 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 |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9804 = 2^{2} \cdot 3 \cdot 19 \cdot 43 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 9754 & 9795 \end{array}\right),\left(\begin{array}{rr} 9793 & 12 \\ 9792 & 13 \end{array}\right),\left(\begin{array}{rr} 6394 & 3 \\ 4305 & 9796 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 4095 & 820 \\ 4058 & 809 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 8266 & 3 \\ 5133 & 9796 \end{array}\right),\left(\begin{array}{rr} 6535 & 9792 \\ 6530 & 9731 \end{array}\right)$.
The torsion field $K:=\Q(E[9804])$ is a degree-$19723753082880$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9804\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 7353 = 3^{2} \cdot 19 \cdot 43 \) |
| $3$ | additive | $8$ | \( 19 \) |
| $19$ | split multiplicative | $20$ | \( 774 = 2 \cdot 3^{2} \cdot 43 \) |
| $43$ | split multiplicative | $44$ | \( 342 = 2 \cdot 3^{2} \cdot 19 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 14706.q
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 4902.f1, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{817}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | 4.0.7353.2 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.23085974187.5 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.36088866774801.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $9$ | 9.3.756526128527708090188992.3 | \(\Z/18\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.112673357446242804568580584050557685138332061490392278372352.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | ss | ord | ss | ord | ord | split | ss | ord | ord | ord | ss | split | ss |
| $\lambda$-invariant(s) | 8 | - | 1,1 | 1 | 1,1 | 1 | 1 | 2 | 3,1 | 1 | 1 | 1 | 1,1 | 2 | 1,1 |
| $\mu$-invariant(s) | 0 | - | 0,0 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.