Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-10321633x-8026548863\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-10321633xz^2-8026548863z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-836052300x-5853862278000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2643, 28000)$ | $1.2814556571819633385796245275$ | $\infty$ |
| $(-2713, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2713, 0\right) \), \((-2643,\pm 28000)\), \((41387,\pm 8393700)\), \((165357,\pm 67228000)\)
Invariants
| Conductor: | $N$ | = | \( 33600 \) | = | $2^{6} \cdot 3 \cdot 5^{2} \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $42520434281472000000000$ | = | $2^{19} \cdot 3 \cdot 5^{9} \cdot 7^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{29689921233686449}{10380965400750} \) | = | $2^{-1} \cdot 3^{-1} \cdot 5^{-3} \cdot 7^{-12} \cdot 23^{3} \cdot 13463^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0434657400928929102521704495$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1990260130359247588259426007$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0394034516317685$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.762928361048434$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2814556571819633385796245275$ |
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| Real period: | $\Omega$ | ≈ | $0.086645238355003269746515687524$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2^{2}\cdot1\cdot2^{2}\cdot( 2^{2} \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.3295374811791314134371142300 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.329537481 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.086645 \cdot 1.281456 \cdot 192}{2^2} \\ & \approx 5.329537481\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2654208 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{9}^{*}$ | additive | -1 | 6 | 19 | 1 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $7$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
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 | 4.6.0.1 |
| $3$ | 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 $384$, genus $5$, and generators
$\left(\begin{array}{rr} 488 & 837 \\ 123 & 86 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 798 & 641 \\ 523 & 28 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 374 & 11 \end{array}\right),\left(\begin{array}{rr} 194 & 819 \\ 495 & 374 \end{array}\right),\left(\begin{array}{rr} 241 & 24 \\ 372 & 289 \end{array}\right),\left(\begin{array}{rr} 576 & 719 \\ 305 & 546 \end{array}\right),\left(\begin{array}{rr} 817 & 24 \\ 816 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ 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 | $4$ | \( 75 = 3 \cdot 5^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 1344 = 2^{6} \cdot 3 \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 33600ex
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210a7, its twist by $-40$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-6})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $6$ | 6.0.15116544000.27 | \(\Z/6\Z\) | not in database |
| $8$ | 8.4.47775744000000.32 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.226586787840000.109 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.448084224000000.14 | \(\Z/8\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.6.560293142487577170599106772992000000000000000.1 | \(\Z/2\Z \oplus \Z/18\Z\) | 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 | add | nonsplit | add | split | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | - | 2 | 1,1 | 1 | 1 | 1 | 1,3 | 3 | 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.