Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-454955x+118072977\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-454955xz^2+118072977z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-589621707x+5510581680006\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{10}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(418, 781\right) \) | $2.7072254601651052675940761246$ | $\infty$ |
| \( \left(364, 673\right) \) | $0$ | $10$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([418:781:1]\) | $2.7072254601651052675940761246$ | $\infty$ |
| \([364:673:1]\) | $0$ | $10$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(15051, 213840\right) \) | $2.7072254601651052675940761246$ | $\infty$ |
| \( \left(13107, 184680\right) \) | $0$ | $10$ |
Integral points
\( \left(-776, 2383\right) \), \( \left(-776, -1607\right) \), \( \left(-206, 14353\right) \), \( \left(-206, -14147\right) \), \( \left(364, 673\right) \), \( \left(364, -1037\right) \), \( \left(394, -47\right) \), \( \left(394, -347\right) \), \( \left(418, 781\right) \), \( \left(418, -1199\right) \), \( \left(544, 5353\right) \), \( \left(544, -5897\right) \), \( \left(1444, 49003\right) \), \( \left(1444, -50447\right) \), \( \left(1794, 70353\right) \), \( \left(1794, -72147\right) \)
\([-776:2383:1]\), \([-776:-1607:1]\), \([-206:14353:1]\), \([-206:-14147:1]\), \([364:673:1]\), \([364:-1037:1]\), \([394:-47:1]\), \([394:-347:1]\), \([418:781:1]\), \([418:-1199:1]\), \([544:5353:1]\), \([544:-5897:1]\), \([1444:49003:1]\), \([1444:-50447:1]\), \([1794:70353:1]\), \([1794:-72147:1]\)
\((-27933,\pm 430920)\), \((-7413,\pm 3078000)\), \((13107,\pm 184680)\), \((14187,\pm 32400)\), \((15051,\pm 213840)\), \((19587,\pm 1215000)\), \((51987,\pm 10740600)\), \((64587,\pm 15390000)\)
Invariants
| Conductor: | $N$ | = | \( 6270 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 19$ |
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| Minimal Discriminant: | $\Delta$ | = | $301547812500000$ | = | $2^{5} \cdot 3^{5} \cdot 5^{10} \cdot 11 \cdot 19^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{10414276373665867414321}{301547812500000} \) | = | $2^{-5} \cdot 3^{-5} \cdot 5^{-10} \cdot 11^{-1} \cdot 19^{-2} \cdot 29^{3} \cdot 89^{3} \cdot 8461^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8788362897524418246402123914$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8788362897524418246402123914$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9942357097970014$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.798282281149496$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.7072254601651052675940761246$ |
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| Real period: | $\Omega$ | ≈ | $0.50782646680532685182412406365$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 500 $ = $ 5\cdot5\cdot( 2 \cdot 5 )\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $10$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.8740037014053527078658190679 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.874003701 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.507826 \cdot 2.707225 \cdot 500}{10^2} \\ & \approx 6.874003701\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 64000 |
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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 semistable. There are 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 2.3.0.1 | $3$ |
| $5$ | 5B.1.1 | 5.24.0.1 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 25080 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 25061 & 20 \\ 25060 & 21 \end{array}\right),\left(\begin{array}{rr} 16 & 5 \\ 12495 & 25066 \end{array}\right),\left(\begin{array}{rr} 20536 & 5 \\ 15915 & 25066 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 24840 & 24731 \end{array}\right),\left(\begin{array}{rr} 15051 & 20 \\ 190 & 1267 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 6271 & 20 \\ 12550 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 16736 & 5 \\ 25035 & 25066 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22441 & 20 \\ 23770 & 201 \end{array}\right)$.
The torsion field $K:=\Q(E[25080])$ is a degree-$199702609920000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/25080\Z)$.
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 | $4$ | \( 33 = 3 \cdot 11 \) |
| $3$ | split multiplicative | $4$ | \( 2090 = 2 \cdot 5 \cdot 11 \cdot 19 \) |
| $5$ | split multiplicative | $6$ | \( 209 = 11 \cdot 19 \) |
| $11$ | split multiplicative | $12$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 6270r
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
This elliptic curve is its own minimal quadratic twist.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{10}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{66}) \) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $4$ | \(\Q(\sqrt{31 + \sqrt{-95}})\) | \(\Z/20\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $8$ | deg 8 | \(\Z/30\Z\) | not in database |
| $16$ | deg 16 | \(\Z/40\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/30\Z\) | not in database |
| $20$ | 20.0.404474030606051350893885493549332539093017578125.4 | \(\Z/5\Z \oplus \Z/10\Z\) | 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 | split | ord | split | ord | ord | nonsplit | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 2 | 2 | 1 | 2 | 1 | 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 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.