Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+731508x+10909262576\)
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(homogenize, simplify) |
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\(y^2z=x^3+731508xz^2+10909262576z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+731508x+10909262576\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1858, 56000)$ | $0.89494940331649729518440028616$ | $\infty$ |
| $(-2108, 0)$ | $0$ | $2$ |
Integral points
\( \left(-2108, 0\right) \), \((-1858,\pm 56000)\), \((-1208,\pm 90900)\), \((13517,\pm 1578125)\), \((54142,\pm 12600000)\)
Invariants
| Conductor: | $N$ | = | \( 20160 \) | = | $2^{6} \cdot 3^{2} \cdot 5 \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $-51438240000000000000000$ | = | $-1 \cdot 2^{20} \cdot 3^{8} \cdot 5^{16} \cdot 7^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{226523624554079}{269165039062500} \) | = | $2^{-2} \cdot 3^{-2} \cdot 5^{-16} \cdot 7^{-2} \cdot 47^{3} \cdot 1297^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.0368827417895632020678929873$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4478558266155903922444221867$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1083142308531486$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.028267017300838$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.89494940331649729518440028616$ |
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| Real period: | $\Omega$ | ≈ | $0.087957680906410141337720124459$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 256 $ = $ 2^{2}\cdot2\cdot2^{4}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.0379311388348558694982160760 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.037931139 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.087958 \cdot 0.894949 \cdot 256}{2^2} \\ & \approx 5.037931139\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1572864 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{10}^{*}$ | additive | 1 | 6 | 20 | 2 |
| $3$ | $2$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $16$ | $I_{16}$ | split multiplicative | -1 | 1 | 16 | 16 |
| $7$ | $2$ | $I_{2}$ | nonsplit 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 | 16.96.0.191 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 2017 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3359 & 3352 \\ 0 & 2939 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 25 & 16 \\ 1624 & 2249 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 16 \\ 2775 & 793 \end{array}\right),\left(\begin{array}{rr} 985 & 16 \\ 1974 & 2473 \end{array}\right),\left(\begin{array}{rr} 1091 & 3336 \\ 3234 & 475 \end{array}\right),\left(\begin{array}{rr} 3329 & 32 \\ 3328 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 4 & 129 \end{array}\right)$.
The torsion field $K:=\Q(E[3360])$ is a degree-$23781703680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3360\Z)$.
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 | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 2240 = 2^{6} \cdot 5 \cdot 7 \) |
| $5$ | split multiplicative | $6$ | \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 20160.cz
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 210.e8, its twist by $-24$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{6})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{14})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{-14})\) | \(\Z/8\Z\) | not in database |
| $4$ | 4.0.55296.2 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.12745506816.1 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.12230590464.4 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | 8.0.29365647704064.50 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/32\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | nonsplit | ord | ord | ord | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | - | 2 | 1 | 1 | 1 | 1 | 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 | 0,0 |
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.