Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-159608x+888516288\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-159608xz^2+888516288z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-12928275x+647767158750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(208, 29400)$ | $0.61571359426171927303675857901$ | $\infty$ |
| $(-1017, 0)$ | $0$ | $2$ |
Integral points
\( \left(-1017, 0\right) \), \((-728,\pm 24888)\), \((208,\pm 29400)\), \((1384,\pm 57624)\), \((3883,\pm 242550)\)
Invariants
| Conductor: | $N$ | = | \( 79800 \) | = | $2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $-340827856037424000000$ | = | $-1 \cdot 2^{10} \cdot 3^{4} \cdot 5^{6} \cdot 7^{12} \cdot 19 $ |
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| j-invariant: | $j$ | = | \( -\frac{28104147578308}{21301741002339} \) | = | $-1 \cdot 2^{2} \cdot 3^{-4} \cdot 7^{-12} \cdot 19^{-1} \cdot 107^{3} \cdot 179^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.6188730288695313922534668987$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2365314221858601137720604642$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0526082543070654$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.84912204397065$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.61571359426171927303675857901$ |
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| Real period: | $\Omega$ | ≈ | $0.13807744234734567564405972168$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 384 $ = $ 2\cdot2^{2}\cdot2^{2}\cdot( 2^{2} \cdot 3 )\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.1615511981583548939281149061 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.161551198 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.138077 \cdot 0.615714 \cdot 384}{2^2} \\ & \approx 8.161551198\end{aligned}$$
Modular invariants
Modular form 79800.2.a.bm
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3145728 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | -1 | 3 | 10 | 0 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $4$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $19$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 5320 = 2^{3} \cdot 5 \cdot 7 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 4036 & 1065 \\ 4055 & 6 \end{array}\right),\left(\begin{array}{rr} 2531 & 2530 \\ 1210 & 4931 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 2531 & 400 \\ 5210 & 4661 \end{array}\right),\left(\begin{array}{rr} 5313 & 8 \\ 5312 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 5314 & 5315 \end{array}\right),\left(\begin{array}{rr} 3041 & 3200 \\ 460 & 2161 \end{array}\right),\left(\begin{array}{rr} 1063 & 0 \\ 0 & 5319 \end{array}\right)$.
The torsion field $K:=\Q(E[5320])$ is a degree-$3812504371200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/5320\Z)$.
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 | $2$ | \( 475 = 5^{2} \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \) |
| $5$ | additive | $14$ | \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \) |
| $7$ | split multiplicative | $8$ | \( 11400 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 4200 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 79800bu
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 3192c4, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{95}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.120437455360000.14 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.2218908160000.5 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | split | ord | ord | ord | split | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 2 | - | 2 | 1 | 1 | 1 | 2 | 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 |
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.