Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-18479x-950282\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-18479xz^2-950282z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-295659x-61113690\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{13121}{25}, \frac{1412579}{125}\right) \) | $6.6761715387439359769573685314$ | $\infty$ |
| \( \left(-71, 35\right) \) | $0$ | $2$ |
| \( \left(157, -79\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([65605:1412579:125]\) | $6.6761715387439359769573685314$ | $\infty$ |
| \([-71:35:1]\) | $0$ | $2$ |
| \([157:-79:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{52459}{25}, \frac{11563552}{125}\right) \) | $6.6761715387439359769573685314$ | $\infty$ |
| \( \left(-285, 0\right) \) | $0$ | $2$ |
| \( \left(627, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-71, 35\right) \), \( \left(157, -79\right) \)
\([-71:35:1]\), \([157:-79:1]\)
\( \left(-285, 0\right) \), \( \left(627, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 55233 \) | = | $3^{2} \cdot 17 \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $9911673254961$ | = | $3^{6} \cdot 17^{2} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{20346417}{289} \) | = | $3^{3} \cdot 7^{3} \cdot 13^{3} \cdot 17^{-2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2983159226959589400360872694$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.72320971122131613566604906501$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0296285559493121$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.7627561511294743$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.6761715387439359769573685314$ |
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| Real period: | $\Omega$ | ≈ | $0.40983143850773445583844964131$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4722099708956446370493411194 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.472209971 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.409831 \cdot 6.676172 \cdot 32}{4^2} \\ & \approx 5.472209971\end{aligned}$$
Modular invariants
Modular form 55233.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 110592 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $19$ | $4$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 |
|---|---|---|---|
| $2$ | 2Cs | 16.48.0.20 | $48$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 31008 = 2^{5} \cdot 3 \cdot 17 \cdot 19 \), index $1536$, genus $53$, and generators
$\left(\begin{array}{rr} 9 & 32 \\ 30952 & 30809 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 30977 & 32 \\ 30976 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 20671 & 0 \\ 0 & 31007 \end{array}\right),\left(\begin{array}{rr} 685 & 19608 \\ 10602 & 24853 \end{array}\right),\left(\begin{array}{rr} 13567 & 26790 \\ 15504 & 1 \end{array}\right),\left(\begin{array}{rr} 29 & 8 \\ 4252 & 1173 \end{array}\right),\left(\begin{array}{rr} 16319 & 0 \\ 0 & 31007 \end{array}\right),\left(\begin{array}{rr} 5245 & 6612 \\ 26904 & 13225 \end{array}\right)$.
The torsion field $K:=\Q(E[31008])$ is a degree-$118514421596160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/31008\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
| $3$ | additive | $6$ | \( 6137 = 17 \cdot 19^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 153 = 3^{2} \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 55233g
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 17a2, its twist by $57$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(i, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{-57})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.225701826437376.36 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.23804489507067.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | ord | add | ord | ord | ss | ord | nonsplit | add | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 4 | - | 1 | 1 | 1,1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-invariant(s) | 1 | - | 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.