Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1262579938x-18313477533969\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1262579938xz^2-18313477533969z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1636303599675x-854409063270854250\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 4550 \) | = | $2 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-16065240590756033331200000000$ | = | $-1 \cdot 2^{63} \cdot 5^{8} \cdot 7^{3} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{14245586655234650511684983641}{1028175397808386133196800} \) | = | $-1 \cdot 2^{-63} \cdot 5^{-2} \cdot 7^{-3} \cdot 13^{-1} \cdot 61^{3} \cdot 97^{3} \cdot 409693^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $4.1618667336888665824489290099$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $3.3571477774718163951485493433$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0476037319513174$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $8.856873492265683$ | |||
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: | $\Omega$ | ≈ | $0.012611147043990680367976301879$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 126 $ = $ ( 3^{2} \cdot 7 )\cdot2\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.5890045275428257263650140367 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.589004528 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.012611 \cdot 1.000000 \cdot 126}{1^2} \\ & \approx 1.589004528\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4572288 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $63$ | $I_{63}$ | split multiplicative | -1 | 1 | 63 | 63 |
| $5$ | $2$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 2 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $13$ | $1$ | $I_{1}$ | nonsplit 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 | 9.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 32760 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 32743 & 18 \\ 32742 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 20671 & 6570 \\ 22545 & 25861 \end{array}\right),\left(\begin{array}{rr} 13103 & 0 \\ 0 & 32759 \end{array}\right),\left(\begin{array}{rr} 8191 & 6570 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 24571 & 22950 \\ 0 & 17291 \end{array}\right),\left(\begin{array}{rr} 18721 & 6570 \\ 24345 & 26371 \end{array}\right),\left(\begin{array}{rr} 16381 & 6570 \\ 3285 & 26371 \end{array}\right)$.
The torsion field $K:=\Q(E[32760])$ is a degree-$1051769626951680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/32760\Z)$.
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 | $4$ | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| $3$ | good | $2$ | \( 325 = 5^{2} \cdot 13 \) |
| $5$ | additive | $18$ | \( 182 = 2 \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 325 = 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 4550.r
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 910.e1, its twist by $5$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.728.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.385828352.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.803278125.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.70270770375.5 | \(\Z/9\Z\) | not in database |
| $6$ | 6.0.1788696000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.2.2701547996147423119903752000000000000000.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.1808585359011771461918813778234729984000000000.1 | \(\Z/18\Z\) | 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 | ord | add | nonsplit | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 0 | - | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | 2 | - | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.