Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-544300x+155242000\)
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(homogenize, simplify) |
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\(y^2z=x^3-544300xz^2+155242000z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-544300x+155242000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(330, 3400)$ | $0.40597704336204157169421404825$ | $\infty$ |
| $(160, 8500)$ | $0.47178252834707602852477172028$ | $\infty$ |
Integral points
\((-715,\pm 13375)\), \((-420,\pm 17600)\), \((-214,\pm 16184)\), \((160,\pm 8500)\), \((330,\pm 3400)\), \((364,\pm 2312)\), \((410,\pm 1000)\), \((426,\pm 824)\), \((874,\pm 18632)\), \((1180,\pm 34000)\), \((2285,\pm 104125)\), \((4410,\pm 289000)\), \((12160,\pm 1338500)\), \((27530,\pm 4566200)\)
Invariants
| Conductor: | $N$ | = | \( 27200 \) | = | $2^{6} \cdot 5^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-90870848000000000$ | = | $-1 \cdot 2^{15} \cdot 5^{9} \cdot 17^{5} $ |
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| j-invariant: | $j$ | = | \( -\frac{34831225434312}{177482125} \) | = | $-1 \cdot 2^{3} \cdot 3^{3} \cdot 5^{-3} \cdot 17^{-5} \cdot 5443^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1006341592821503140893936842$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.42948122736516849001747386577$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9892295652888757$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.018536508260937$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.17356588666794539642931731278$ |
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| Real period: | $\Omega$ | ≈ | $0.34092961091774630191898823510$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ 2^{2}\cdot2^{2}\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $4.7339000168237019307791723774 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.733900017 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.340930 \cdot 0.173566 \cdot 80}{1^2} \\ & \approx 4.733900017\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 599040 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | 1 | 6 | 15 | 0 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $17$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 680 = 2^{3} \cdot 5 \cdot 17 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 679 & 0 \end{array}\right),\left(\begin{array}{rr} 241 & 2 \\ 241 & 3 \end{array}\right),\left(\begin{array}{rr} 341 & 2 \\ 341 & 3 \end{array}\right),\left(\begin{array}{rr} 511 & 2 \\ 511 & 3 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 137 & 2 \\ 137 & 3 \end{array}\right),\left(\begin{array}{rr} 679 & 2 \\ 678 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[680])$ is a degree-$28877783040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/680\Z)$.
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 | $4$ | \( 425 = 5^{2} \cdot 17 \) |
| $5$ | additive | $18$ | \( 64 = 2^{6} \) |
| $17$ | split multiplicative | $18$ | \( 1600 = 2^{6} \cdot 5^{2} \) |
Isogenies
This curve has no rational isogenies. Its isogeny class 27200.c consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 2720.a1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.680.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.314432000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | ss | add | ord | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 6,4 | - | 2 | 4 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.