Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-60216x+1821384\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-60216xz^2+1821384z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-78040611x+86149097694\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-55, 2257\right) \) | $1.4450805062030174953190441274$ | $\infty$ |
| \( \left(\frac{123}{4}, -\frac{123}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-55:2257:1]\) | $1.4450805062030174953190441274$ | $\infty$ |
| \([246:-123:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-1965, 481572\right) \) | $1.4450805062030174953190441274$ | $\infty$ |
| \( \left(1122, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-55, 2257\right) \), \( \left(-55, -2202\right) \), \( \left(241, 1054\right) \), \( \left(241, -1295\right) \), \( \left(699, 17025\right) \), \( \left(699, -17724\right) \)
\([-55:2257:1]\), \([-55:-2202:1]\), \([241:1054:1]\), \([241:-1295:1]\), \([699:17025:1]\), \([699:-17724:1]\)
\((-1965,\pm 481572)\), \((8691,\pm 253692)\), \((25179,\pm 3752892)\)
Invariants
| Conductor: | $N$ | = | \( 66066 \) | = | $2 \cdot 3 \cdot 7 \cdot 11^{2} \cdot 13$ |
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| Minimal Discriminant: | $\Delta$ | = | $12501310278569112$ | = | $2^{3} \cdot 3^{10} \cdot 7^{6} \cdot 11^{3} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{18142312145028947}{9392419442952} \) | = | $2^{-3} \cdot 3^{-10} \cdot 7^{-6} \cdot 13^{-2} \cdot 127^{3} \cdot 2069^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7807294148847337502026700852$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1812555966851411141871841907$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9963520222537113$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.021360810770683$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.4450805062030174953190441274$ |
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| Real period: | $\Omega$ | ≈ | $0.35227247188048906836880273072$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.0362483279457815574275333527 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.036248328 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.352272 \cdot 1.445081 \cdot 16}{2^2} \\ & \approx 2.036248328\end{aligned}$$
Modular invariants
Modular form 66066.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 691200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $7$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $11$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $13$ | $2$ | $I_{2}$ | 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 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 923 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1845 & 4 \\ 1844 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1156 & 697 \\ 1617 & 232 \end{array}\right),\left(\begin{array}{rr} 617 & 4 \\ 1234 & 9 \end{array}\right),\left(\begin{array}{rr} 1585 & 4 \\ 1322 & 9 \end{array}\right),\left(\begin{array}{rr} 844 & 1 \\ 1175 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[1848])$ is a degree-$163499212800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.
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 | $4$ | \( 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1573 = 11^{2} \cdot 13 \) |
| $5$ | good | $2$ | \( 22022 = 2 \cdot 7 \cdot 11^{2} \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 9438 = 2 \cdot 3 \cdot 11^{2} \cdot 13 \) |
| $11$ | additive | $42$ | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 66066.a
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{22}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.18783072.3 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.22579442800459776.57 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | ord | nonsplit | add | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 5 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 3 | 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.