Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1079838x-432138469\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1079838xz^2-432138469z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1399470075x-20140860350250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(56289, 13324447)$ | $4.8400797621263100908977349199$ | $\infty$ |
| $(-2445/4, 2441/8)$ | $0$ | $2$ |
Integral points
\( \left(56289, 13324447\right) \), \( \left(56289, -13380737\right) \)
Invariants
| Conductor: | $N$ | = | \( 10450 \) | = | $2 \cdot 5^{2} \cdot 11 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $79941690125000000$ | = | $2^{6} \cdot 5^{9} \cdot 11^{6} \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{8912089320684236569}{5116268168000} \) | = | $2^{-6} \cdot 5^{-3} \cdot 11^{-6} \cdot 19^{-2} \cdot 23^{3} \cdot 109^{3} \cdot 827^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1893339933521511661197867205$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3846150371351009788194070539$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9673933417704673$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.758429934351057$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8400797621263100908977349199$ |
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| Real period: | $\Omega$ | ≈ | $0.14810675463904050021221567764$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ ( 2 \cdot 3 )\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.6021820691515230137983304218 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.602182069 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.148107 \cdot 4.840080 \cdot 48}{2^2} \\ & \approx 8.602182069\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 193536 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $11$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $19$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 \( 12540 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9901 & 12 \\ 9246 & 73 \end{array}\right),\left(\begin{array}{rr} 4181 & 12 \\ 4180 & 1 \end{array}\right),\left(\begin{array}{rr} 12529 & 12 \\ 12528 & 13 \end{array}\right),\left(\begin{array}{rr} 9121 & 12 \\ 4566 & 73 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 12490 & 12531 \end{array}\right),\left(\begin{array}{rr} 12530 & 12537 \\ 10059 & 8 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 5235 & 1048 \\ 5198 & 1037 \end{array}\right)$.
The torsion field $K:=\Q(E[12540])$ is a degree-$37444239360000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12540\Z)$.
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 | $4$ | \( 25 = 5^{2} \) |
| $3$ | good | $2$ | \( 475 = 5^{2} \cdot 19 \) |
| $5$ | additive | $18$ | \( 418 = 2 \cdot 11 \cdot 19 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 10450y
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2090d4, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.3494480.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.146611125.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.305284761760000.18 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.24728065702560000.5 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 |
| $18$ | 18.0.50807160047406080794189709683036713375000000000000.1 | \(\Z/18\Z\) | 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 | ord | add | ord | nonsplit | ord | ss | split | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 3 | - | 1 | 1 | 1 | 1,1 | 2 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 1 | - | 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.