Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-4566x+119916\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-4566xz^2+119916z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-73059x+7601566\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(39, -15\right) \) | $1.6460586798528164781941450636$ | $\infty$ |
| \( \left(37, 6\right) \) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([39:-15:1]\) | $1.6460586798528164781941450636$ | $\infty$ |
| \([37:6:1]\) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(155, 36\right) \) | $1.6460586798528164781941450636$ | $\infty$ |
| \( \left(147, 196\right) \) | $0$ | $3$ |
Integral points
\( \left(37, 6\right) \), \( \left(37, -43\right) \), \( \left(39, -15\right) \), \( \left(39, -24\right) \), \( \left(135, 1329\right) \), \( \left(135, -1464\right) \)
\([37:6:1]\), \([37:-43:1]\), \([39:-15:1]\), \([39:-24:1]\), \([135:1329:1]\), \([135:-1464:1]\)
\((147,\pm 196)\), \((155,\pm 36)\), \((539,\pm 11172)\)
Invariants
| Conductor: | $N$ | = | \( 882 \) | = | $2 \cdot 3^{2} \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1245197016$ | = | $-1 \cdot 2^{3} \cdot 3^{3} \cdot 7^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{67645179}{8} \) | = | $-1 \cdot 2^{-3} \cdot 3^{3} \cdot 7 \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.77073429669175094947238060706$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.80119220817881867677999919780$ |
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| $abc$ quality: | $Q$ | ≈ | $1.032612370203439$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.439702789238773$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.6460586798528164781941450636$ |
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| Real period: | $\Omega$ | ≈ | $1.4743501869071501600900180928$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $1.6179112815340912147969660175 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 1.617911282 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.474350 \cdot 1.646059 \cdot 6}{3^2} \\ & \approx 1.617911282\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1008 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $7$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 0 |
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 | 9.24.0.2 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 139 & 486 \\ 180 & 115 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 12 & 217 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 127 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 487 & 18 \\ 486 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 253 & 18 \\ 261 & 163 \end{array}\right),\left(\begin{array}{rr} 383 & 270 \\ 171 & 445 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 440 & 445 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$83607552$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
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 | $4$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | additive | $6$ | \( 49 = 7^{2} \) |
| $7$ | additive | $26$ | \( 18 = 2 \cdot 3^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 882.a
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 882.e1, its twist by $-7$.
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 |
|---|---|---|---|
| $3$ | 3.1.1176.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.33191424.2 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.5250987.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.1688134559643.1 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.8549394874383196572862347.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.37954454265953846507077632.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | add | ord | add | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord | ord |
| $\lambda$-invariant(s) | 2 | - | 1 | - | 1 | 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 |
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.