Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-14646488x-21340722733\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-14646488xz^2-21340722733z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-234343803x-1366040598698\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-9573/4, 9569/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 16830 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $4273611826236328125000$ | = | $2^{3} \cdot 3^{9} \cdot 5^{12} \cdot 11^{3} \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{476646772170172569823801}{5862293314453125000} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{-12} \cdot 11^{-3} \cdot 17^{-4} \cdot 23^{3} \cdot 107^{3} \cdot 31741^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9607874489120620788144189750$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4114813045780072331167963565$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0086073801975677$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.280264059186998$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.077230561206391838428001414314$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 3\cdot2^{2}\cdot2\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.8535334689534041222720339435 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.853533469 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.077231 \cdot 1.000000 \cdot 96}{2^2} \\ & \approx 1.853533469\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1769472 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $11$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $17$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | 8.12.0.7 |
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 22440 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 17 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 14521 & 24 \\ 17172 & 289 \end{array}\right),\left(\begin{array}{rr} 16831 & 24 \\ 16830 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 21134 & 13211 \end{array}\right),\left(\begin{array}{rr} 12162 & 15919 \\ 7037 & 11972 \end{array}\right),\left(\begin{array}{rr} 2056 & 3 \\ 9741 & 22354 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22417 & 24 \\ 22416 & 25 \end{array}\right),\left(\begin{array}{rr} 22424 & 11199 \\ 18985 & 15334 \end{array}\right),\left(\begin{array}{rr} 8977 & 24 \\ 17964 & 289 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[22440])$ is a degree-$95296684032000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/22440\Z)$.
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 | $4$ | \( 99 = 3^{2} \cdot 11 \) |
| $3$ | additive | $2$ | \( 17 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 3366 = 2 \cdot 3^{2} \cdot 11 \cdot 17 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 1530 = 2 \cdot 3^{2} \cdot 5 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 16830cd
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 5610t5, its twist by $-3$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{66}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-33}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | not in database |
| $3$ | 3.1.867.1 | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-33})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-22})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{11})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-3})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.2255067.2 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.1536765018624.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.192095627328.4 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.384864768.1 | \(\Z/12\Z\) | not in database |
| $8$ | 8.4.5416809268248576.43 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.77720518656.6 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.29712900642147807147732528143704650293701171875.2 | \(\Z/18\Z\) | 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 | 11 | 17 |
|---|---|---|---|---|---|
| Reduction type | split | add | nonsplit | nonsplit | split |
| $\lambda$-invariant(s) | 6 | - | 0 | 0 | 1 |
| $\mu$-invariant(s) | 1 | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.