Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-657608x+246220788\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-657608xz^2+246220788z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-53266275x+179654753250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(268, 9450\right) \) | $3.7734216292048191574505882203$ | $\infty$ |
| \( \left(-957, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([268:9450:1]\) | $3.7734216292048191574505882203$ | $\infty$ |
| \([-957:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(2415, 255150\right) \) | $3.7734216292048191574505882203$ | $\infty$ |
| \( \left(-8610, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-957, 0\right) \), \((268,\pm 9450)\)
\([-957:0:1]\), \([268:\pm 9450:1]\)
\( \left(-957, 0\right) \), \((268,\pm 9450)\)
Invariants
| Conductor: | $N$ | = | \( 20400 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 17$ |
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| Minimal Discriminant: | $\Delta$ | = | $-8036072572032000000$ | = | $-1 \cdot 2^{13} \cdot 3^{2} \cdot 5^{6} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{491411892194497}{125563633938} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-2} \cdot 17^{-8} \cdot 23^{3} \cdot 47^{3} \cdot 73^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.3449444224184677509554388448$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.84707828564147225423782705673$ |
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| $abc$ quality: | $Q$ | ≈ | $1.036242441540377$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.257168010334941$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.7734216292048191574505882203$ |
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| Real period: | $\Omega$ | ≈ | $0.22214929700322732808846786275$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7061036977969865156267629020 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.706103698 \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 0.222149 \cdot 3.773422 \cdot 32}{2^2} \\ & \approx 6.706103698\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 393216 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.48.0.224 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1360 = 2^{4} \cdot 5 \cdot 17 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 15 & 2 \\ 1262 & 1347 \end{array}\right),\left(\begin{array}{rr} 356 & 885 \\ 295 & 146 \end{array}\right),\left(\begin{array}{rr} 543 & 0 \\ 0 & 1359 \end{array}\right),\left(\begin{array}{rr} 434 & 475 \\ 295 & 1214 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1345 & 16 \\ 1344 & 17 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1356 & 1357 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 241 & 560 \\ 840 & 401 \end{array}\right)$.
The torsion field $K:=\Q(E[1360])$ is a degree-$4812963840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1360\Z)$.
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 | $4$ | \( 25 = 5^{2} \) |
| $3$ | split multiplicative | $4$ | \( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \) |
| $5$ | additive | $14$ | \( 816 = 2^{4} \cdot 3 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 20400cx
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 102b6, its twist by $-20$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-10}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{5})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{5})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.849346560000.16 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.849346560000.17 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.40960000.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | 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 | \(\Z/12\Z\) | 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 | split | add | ss | ord | ord | nonsplit | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 2 | - | 1,1 | 1 | 1 | 5 | 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 | 0,0 |
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.