Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-10065x-389499\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-10065xz^2-389499z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-13044267x-18133332570\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-58, 29\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-58:29:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-2085, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-58, 29\right) \)
\([-58:29:1]\)
\( \left(-2085, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 66 \) | = | $2 \cdot 3 \cdot 11$ |
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| Minimal Discriminant: | $\Delta$ | = | $1932612$ | = | $2^{2} \cdot 3 \cdot 11^{5} $ |
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| j-invariant: | $j$ | = | \( \frac{112763292123580561}{1932612} \) | = | $2^{-2} \cdot 3^{-1} \cdot 11^{-5} \cdot 179^{3} \cdot 2699^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.74785326639654765811527565409$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.74785326639654765811527565409$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0637869122465706$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $9.371671336713842$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.47664592624499408526241012510$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 10 $ = $ 2\cdot1\cdot5 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.1916148156124852131560253128 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.191614816 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.476646 \cdot 1.000000 \cdot 10}{2^2} \\ & \approx 1.191614816\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 100 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
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 | 8.6.0.4 | $6$ |
| $5$ | 5B.1.2 | 5.24.0.3 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 1301 & 20 \\ 1300 & 21 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 1080 & 971 \end{array}\right),\left(\begin{array}{rr} 991 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1057 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 661 & 20 \\ 10 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 736 & 5 \\ 75 & 1306 \end{array}\right),\left(\begin{array}{rr} 896 & 5 \\ 1275 & 1306 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1320])$ is a degree-$1622016000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1320\Z)$.
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 | $4$ | \( 22 = 2 \cdot 11 \) |
| $5$ | good | $2$ | \( 6 = 2 \cdot 3 \) |
| $11$ | split multiplicative | $12$ | \( 6 = 2 \cdot 3 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 66c
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.2.33.1-132.1-b7 |
| $4$ | \(\Q(\sqrt{1 +4 \sqrt{-2}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\zeta_{5})\) | \(\Z/10\Z\) | not in database |
| $5$ | 5.1.4050000.3 | \(\Z/10\Z\) | not in database |
| $8$ | 8.0.4857532416.8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.1322463200256.6 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.41497747632.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.18530015625.3 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $10$ | 10.2.7924917082500000000.1 | \(\Z/2\Z \oplus \Z/10\Z\) | 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 |
| $16$ | deg 16 | \(\Z/20\Z\) | not in database |
| $20$ | 20.0.33630250781250000000000000000.2 | \(\Z/5\Z \oplus \Z/10\Z\) | not in database |
| $20$ | 20.0.545211086046835507200000000000000000000.1 | \(\Z/20\Z\) | 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 | 11 |
|---|---|---|---|---|
| Reduction type | split | split | ord | split |
| $\lambda$-invariant(s) | 1 | 1 | 2 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 1 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 7$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.