Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-7822813x-2502482383\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-7822813xz^2-2502482383z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-10138365675x-116725402964250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(9182, 831959)$ | $0.16177776258781770798577727393$ | $\infty$ |
| $(-1297/4, 1297/8)$ | $0$ | $2$ |
Integral points
\( \left(-2518, 36359\right) \), \( \left(-2518, -33841\right) \), \( \left(-1972, 73487\right) \), \( \left(-1972, -71515\right) \), \( \left(-1114, 70055\right) \), \( \left(-1114, -68941\right) \), \( \left(-568, 42209\right) \), \( \left(-568, -41641\right) \), \( \left(3098, 53207\right) \), \( \left(3098, -56305\right) \), \( \left(4232, 198359\right) \), \( \left(4232, -202591\right) \), \( \left(4646, 245555\right) \), \( \left(4646, -250201\right) \), \( \left(9182, 831959\right) \), \( \left(9182, -841141\right) \), \( \left(30476, 5282405\right) \), \( \left(30476, -5312881\right) \)
Invariants
| Conductor: | $N$ | = | \( 21450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $27934638157264410000000$ | = | $2^{7} \cdot 3^{14} \cdot 5^{7} \cdot 11^{2} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3388383326345613179401}{1787816842064922240} \) | = | $2^{-7} \cdot 3^{-14} \cdot 5^{-1} \cdot 11^{-2} \cdot 13^{-6} \cdot 1901^{3} \cdot 7901^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9985868894087998852724652359$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1938679331917496979720855693$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0229491064938572$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.938876679902854$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.16177776258781770798577727393$ |
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| Real period: | $\Omega$ | ≈ | $0.095780780700704904983781757717$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2352 $ = $ 7\cdot( 2 \cdot 7 )\cdot2\cdot2\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $9.1111778355965885547900448535 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.111177836 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.095781 \cdot 0.161778 \cdot 2352}{2^2} \\ & \approx 9.111177836\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1806336 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $3$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 521 & 4 \\ 1042 & 9 \end{array}\right),\left(\begin{array}{rr} 314 & 1 \\ 623 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 779 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1081 & 4 \\ 602 & 9 \end{array}\right),\left(\begin{array}{rr} 1557 & 4 \\ 1556 & 5 \end{array}\right),\left(\begin{array}{rr} 977 & 586 \\ 584 & 975 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[1560])$ is a degree-$77290536960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1560\Z)$.
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 | $4$ | \( 25 = 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
| $5$ | additive | $18$ | \( 858 = 2 \cdot 3 \cdot 11 \cdot 13 \) |
| $7$ | good | $2$ | \( 3575 = 5^{2} \cdot 11 \cdot 13 \) |
| $11$ | split multiplicative | $12$ | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 21450.ck
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 4290.h1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.243360.3 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.94758543360000.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.648402306750000.6 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | split | add | ord | split | split | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 3 | 2 | - | 3 | 2 | 2 | 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.