Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-200621x+34570424\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-200621xz^2+34570424z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-260004195x+1613697726366\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(167, 2310\right) \) | $4.4719451306385070481739088016$ | $\infty$ |
| \( \left(252, 55\right) \) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([167:2310:1]\) | $4.4719451306385070481739088016$ | $\infty$ |
| \([252:55:1]\) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(6015, 517104\right) \) | $4.4719451306385070481739088016$ | $\infty$ |
| \( \left(9075, 39204\right) \) | $0$ | $3$ |
Integral points
\( \left(167, 2310\right) \), \( \left(167, -2478\right) \), \( \left(252, 55\right) \), \( \left(252, -308\right) \)
\([167:2310:1]\), \([167:-2478:1]\), \([252:55:1]\), \([252:-308:1]\)
\((6015,\pm 517104)\), \((9075,\pm 39204)\)
Invariants
| Conductor: | $N$ | = | \( 66066 \) | = | $2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $-8426876329872$ | = | $-1 \cdot 2^{4} \cdot 3^{3} \cdot 7 \cdot 11^{8} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{4165894731625}{39312} \) | = | $-1 \cdot 2^{-4} \cdot 3^{-3} \cdot 5^{3} \cdot 7^{-1} \cdot 11 \cdot 13^{-1} \cdot 1447^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6428462390186885009784749262$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.044249390486441471603845874223$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9047764944147519$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.346670627532502$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.4719451306385070481739088016$ |
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| Real period: | $\Omega$ | ≈ | $0.66343198934997710686220705824$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 18 $ = $ 2\cdot3\cdot1\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.9336629085668959796558688375 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.933662909 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.663432 \cdot 4.471945 \cdot 18}{3^2} \\ & \approx 5.933662909\end{aligned}$$
Modular invariants
Modular form 66066.2.a.be
For more coefficients, see the Downloads section to the right.
| Modular degree: | 342144 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1092 = 2^{2} \cdot 3 \cdot 7 \cdot 13 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 925 & 6 \\ 591 & 19 \end{array}\right),\left(\begin{array}{rr} 157 & 6 \\ 471 & 19 \end{array}\right),\left(\begin{array}{rr} 1002 & 97 \\ 367 & 474 \end{array}\right),\left(\begin{array}{rr} 1087 & 6 \\ 1086 & 7 \end{array}\right),\left(\begin{array}{rr} 1089 & 1090 \\ 1082 & 1085 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 547 & 6 \\ 549 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1092])$ is a degree-$15216574464$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1092\Z)$.
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 | $4$ | \( 33033 = 3 \cdot 7 \cdot 11^{2} \cdot 13 \) |
| $3$ | split multiplicative | $4$ | \( 22022 = 2 \cdot 7 \cdot 11^{2} \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $52$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| $13$ | split multiplicative | $14$ | \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 66066bg
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 66066ch1, its twist by $-11$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.132132.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.19065081043008.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.433730593728432.3 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.27890768675857427291952.3 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.172974833066954236468131318330847585653301888155648.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | split | ss | split | add | split | ord | ord | ord | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | 6 | 3,1 | 4 | - | 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 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.