Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-108192x-13662376\)
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(homogenize, simplify) |
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\(y^2z=x^3-108192xz^2-13662376z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-108192x-13662376\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(385, 1323)$ | $2.8230174268670620435470026497$ | $\infty$ |
| $(-182, 0)$ | $0$ | $2$ |
Integral points
\( \left(-182, 0\right) \), \((385,\pm 1323)\)
Invariants
| Conductor: | $N$ | = | \( 141120 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $414970799846400$ | = | $2^{10} \cdot 3^{9} \cdot 5^{2} \cdot 7^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{1594753024}{4725} \) | = | $2^{17} \cdot 3^{-3} \cdot 5^{-2} \cdot 7^{-1} \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6747304058666879240402423588$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.42515346346164466539108339927$ |
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| $abc$ quality: | $Q$ | ≈ | $1.002089692021393$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.9122172371993225$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.8230174268670620435470026497$ |
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| Real period: | $\Omega$ | ≈ | $0.26328621270031961329450316987$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.9460925336546421257732291615 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.946092534 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.263286 \cdot 2.823017 \cdot 32}{2^2} \\ & \approx 5.946092534\end{aligned}$$
Modular invariants
Modular form 141120.2.a.ie
For more coefficients, see the Downloads section to the right.
| Modular degree: | 884736 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 6 | 10 | 0 |
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 1 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 39 & 178 \\ 266 & 401 \end{array}\right),\left(\begin{array}{rr} 530 & 837 \\ 627 & 428 \end{array}\right),\left(\begin{array}{rr} 431 & 828 \\ 432 & 827 \end{array}\right),\left(\begin{array}{rr} 337 & 12 \\ 342 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 829 & 12 \\ 828 & 13 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 419 & 0 \\ 0 & 839 \end{array}\right),\left(\begin{array}{rr} 146 & 11 \\ 129 & 400 \end{array}\right),\left(\begin{array}{rr} 829 & 838 \\ 470 & 429 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 790 & 831 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$743178240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $2$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $2$ | \( 15680 = 2^{6} \cdot 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 28224 = 2^{6} \cdot 3^{2} \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 141120.ie
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 420.c2, its twist by $-168$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-42}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.134400.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{21})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.193616640000.4 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.140519212646400.58 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7965941760000.76 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.31863767040000.47 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.1807281560407928921484478806962995200000000.2 | \(\Z/18\Z\) | 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 | add | add | split | add | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 2 | - | 1 | 1 | 3 | 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 |
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.