Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-8820x+404950\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-8820xz^2+404950z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-141123x+25775678\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(45, 290\right) \) | $1.2473130850258518118287630208$ | $\infty$ |
| \( \left(-\frac{445}{4}, \frac{445}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([45:290:1]\) | $1.2473130850258518118287630208$ | $\infty$ |
| \([-890:445:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(179, 2500\right) \) | $1.2473130850258518118287630208$ | $\infty$ |
| \( \left(-446, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-69, 860\right) \), \( \left(-69, -791\right) \), \( \left(45, 290\right) \), \( \left(45, -335\right) \), \( \left(71, 329\right) \), \( \left(71, -400\right) \)
\([-69:860:1]\), \([-69:-791:1]\), \([45:290:1]\), \([45:-335:1]\), \([71:329:1]\), \([71:-400:1]\)
\((-277,\pm 6604)\), \((179,\pm 2500)\), \((283,\pm 2916)\)
Invariants
| Conductor: | $N$ | = | \( 630 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $-26156861718750$ | = | $-1 \cdot 2 \cdot 3^{14} \cdot 5^{8} \cdot 7 $ |
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| j-invariant: | $j$ | = | \( -\frac{104094944089921}{35880468750} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-8} \cdot 5^{-8} \cdot 7^{-1} \cdot 47041^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2860635738081867991590496665$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.73675742947413195346142704804$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0081875418390414$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.102540019928812$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2473130850258518118287630208$ |
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| Real period: | $\Omega$ | ≈ | $0.63089617802430280164712559433$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.5738501162850242835487787164 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.573850116 \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 0.630896 \cdot 1.247313 \cdot 8}{2^2} \\ & \approx 1.573850116\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2048 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{8}^{*}$ | additive | -1 | 2 | 14 | 8 |
| $5$ | $2$ | $I_{8}$ | nonsplit multiplicative | 1 | 1 | 8 | 8 |
| $7$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.24.0.88 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 559 & 1664 \\ 1112 & 1551 \end{array}\right),\left(\begin{array}{rr} 436 & 421 \\ 999 & 850 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 16 \\ 1016 & 129 \end{array}\right),\left(\begin{array}{rr} 248 & 1 \\ 559 & 10 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1063 & 16 \\ 54 & 745 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 1582 & 1667 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 1676 & 1677 \end{array}\right),\left(\begin{array}{rr} 1665 & 16 \\ 1664 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[1680])$ is a degree-$5945425920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1680\Z)$.
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 | $4$ | \( 63 = 3^{2} \cdot 7 \) |
| $3$ | additive | $8$ | \( 70 = 2 \cdot 5 \cdot 7 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 630.b
consists of 6 curves linked by isogenies of
degrees dividing 8.
Twists
The minimal quadratic twist of this elliptic curve is 210.c3, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/8\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-42}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.39969909374976.80 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.39969909374976.29 | \(\Z/8\Z\) | not in database |
| $8$ | 8.4.650280960000.19 | \(\Z/16\Z\) | not in database |
| $8$ | 8.2.4253299470000.3 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | 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 | nonsplit | add | nonsplit | split | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | - | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-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.