Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-31748152x+59307836224\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-31748152xz^2+59307836224z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-41145605667x+2767683590948574\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(688, 194056)$ | $1.5192889726107699551853114871$ | $\infty$ |
| $(8827/4, -8827/8)$ | $0$ | $2$ |
Integral points
\( \left(688, 194056\right) \), \( \left(688, -194744\right) \), \( \left(13488, 1437256\right) \), \( \left(13488, -1450744\right) \)
Invariants
| Conductor: | $N$ | = | \( 10830 \) | = | $2 \cdot 3 \cdot 5 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $527822323162110000000000$ | = | $2^{10} \cdot 3^{10} \cdot 5^{10} \cdot 19^{7} $ |
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| j-invariant: | $j$ | = | \( \frac{75224183150104868881}{11219310000000000} \) | = | $2^{-10} \cdot 3^{-10} \cdot 5^{-10} \cdot 19^{-1} \cdot 4221361^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2763296543782080245143877450$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.8041101647949877945098740291$ |
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| $abc$ quality: | $Q$ | ≈ | $1.032281228728047$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.828108060599091$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.5192889726107699551853114871$ |
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| Real period: | $\Omega$ | ≈ | $0.088815779865959138253221382299$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 80 $ = $ 2\cdot2\cdot( 2 \cdot 5 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.6987366988835473367844588550 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.698736699 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.088816 \cdot 1.519289 \cdot 80}{2^2} \\ & \approx 2.698736699\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1728000 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $3$ | $2$ | $I_{10}$ | nonsplit multiplicative | 1 | 1 | 10 | 10 |
| $5$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $19$ | $2$ | $I_{1}^{*}$ | additive | -1 | 2 | 7 | 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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 1121 & 20 \\ 1120 & 21 \end{array}\right),\left(\begin{array}{rr} 1014 & 1127 \\ 265 & 184 \end{array}\right),\left(\begin{array}{rr} 761 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 625 & 46 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 11 & 16 \\ 900 & 791 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right),\left(\begin{array}{rr} 231 & 14 \\ 1120 & 1047 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1140])$ is a degree-$945561600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1140\Z)$.
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 | $4$ | \( 361 = 19^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3610 = 2 \cdot 5 \cdot 19^{2} \) |
| $5$ | split multiplicative | $6$ | \( 361 = 19^{2} \) |
| $19$ | additive | $200$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 10830g
consists of 4 curves linked by isogenies of
degrees dividing 10.
Twists
The minimal quadratic twist of this elliptic curve is 570l2, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-19}) \) | \(\Z/10\Z\) | not in database |
| $4$ | 4.0.17100.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{19})\) | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $8$ | 8.4.43357483929600.8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1688960160000.18 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.105560010000.2 | \(\Z/20\Z\) | 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 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/20\Z\) | not in database |
| $16$ | deg 16 | \(\Z/30\Z\) | not in database |
| $20$ | 20.4.3177714547665748337794219970703125.1 | \(\Z/10\Z\) | 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 | split | ord | ord | ord | ord | add | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | 3 | 2 | 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 | 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.