Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-121202x-18202876\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-121202xz^2-18202876z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-157078467x-846917209026\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(5368, 389806)$ | $4.7796286322642526977085795899$ | $\infty$ |
| $(1627/4, -1627/8)$ | $0$ | $2$ |
Integral points
\( \left(5368, 389806\right) \), \( \left(5368, -395174\right) \)
Invariants
| Conductor: | $N$ | = | \( 8330 \) | = | $2 \cdot 5 \cdot 7^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-28397608552810000$ | = | $-1 \cdot 2^{4} \cdot 5^{4} \cdot 7^{6} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1673672305534489}{241375690000} \) | = | $-1 \cdot 2^{-4} \cdot 5^{-4} \cdot 13^{3} \cdot 17^{-6} \cdot 9133^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8876989975156076569287300309$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.91474392298795100437605365918$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9921013018955642$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.200903011707907$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.7796286322642526977085795899$ |
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| Real period: | $\Omega$ | ≈ | $0.12692177369300452869597091217$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2\cdot2^{2}\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.8531115488067859564580997595 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.853111549 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.126922 \cdot 4.779629 \cdot 32}{2^2} \\ & \approx 4.853111549\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 115200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $7$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_{6}$ | nonsplit 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 | 4.6.0.5 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14280 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 6819 & 8176 \\ 7966 & 12363 \end{array}\right),\left(\begin{array}{rr} 1233 & 2044 \\ 10892 & 14225 \end{array}\right),\left(\begin{array}{rr} 10711 & 12264 \\ 8421 & 2185 \end{array}\right),\left(\begin{array}{rr} 8159 & 0 \\ 0 & 14279 \end{array}\right),\left(\begin{array}{rr} 11775 & 4102 \\ 13622 & 4523 \end{array}\right),\left(\begin{array}{rr} 7141 & 12264 \\ 10206 & 2185 \end{array}\right),\left(\begin{array}{rr} 13 & 24 \\ 13572 & 12973 \end{array}\right),\left(\begin{array}{rr} 14257 & 24 \\ 14256 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 16 \\ 314 & 335 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[14280])$ is a degree-$14554402652160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14280\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 49 = 7^{2} \) |
| $3$ | good | $2$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 1666 = 2 \cdot 7^{2} \cdot 17 \) |
| $7$ | additive | $26$ | \( 170 = 2 \cdot 5 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 490 = 2 \cdot 5 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 8330k
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 170b2, its twist by $-7$.
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 |
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/6\Z\) | 2.0.7.1-28900.2-c1 |
| $4$ | 4.2.56644.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{7})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.92610000.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.45474709504.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.51336683776.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3208542736.1 | \(\Z/12\Z\) | not in database |
| $12$ | 12.0.8576612100000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | 16.0.2635455101117021618176.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.1713945250836966664738878300000000.1 | \(\Z/18\Z\) | 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 | nonsplit | ord | split | add | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 5 | 1 | 2 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 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.