Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-232937x+43157829\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-232937xz^2+43157829z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-301887027x+2018099971854\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(313, 861\right) \) | $4.1839591098003210657698858262$ | $\infty$ |
| \( \left(370, 2607\right) \) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([313:861:1]\) | $4.1839591098003210657698858262$ | $\infty$ |
| \([370:2607:1]\) | $0$ | $4$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(11283, 219780\right) \) | $4.1839591098003210657698858262$ | $\infty$ |
| \( \left(13335, 603072\right) \) | $0$ | $4$ |
Integral points
\( \left(313, 861\right) \), \( \left(313, -1174\right) \), \( \left(370, 2607\right) \), \( \left(370, -2977\right) \)
\([313:861:1]\), \([313:-1174:1]\), \([370:2607:1]\), \([370:-2977:1]\)
\((11283,\pm 219780)\), \((13335,\pm 603072)\)
Invariants
| Conductor: | $N$ | = | \( 10470 \) | = | $2 \cdot 3 \cdot 5 \cdot 349$ |
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| Minimal Discriminant: | $\Delta$ | = | $640892891563200$ | = | $2^{6} \cdot 3^{3} \cdot 5^{2} \cdot 349^{4} $ |
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| j-invariant: | $j$ | = | \( \frac{1397790417785316543001}{640892891563200} \) | = | $2^{-6} \cdot 3^{-3} \cdot 5^{-2} \cdot 13^{3} \cdot 349^{-4} \cdot 860077^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7981276278555753094683034823$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7981276278555753094683034823$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9771020392492882$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.260129980160864$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.1839591098003210657698858262$ |
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| Real period: | $\Omega$ | ≈ | $0.50480603092783358241072724838$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.1120877917826519394426941692 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.112087792 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.504806 \cdot 4.183959 \cdot 16}{4^2} \\ & \approx 2.112087792\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 124416 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $349$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | 4.12.0.7 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8376 = 2^{3} \cdot 3 \cdot 349 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3841 & 8 \\ 6988 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 8370 & 8371 \end{array}\right),\left(\begin{array}{rr} 3144 & 7337 \\ 3169 & 3216 \end{array}\right),\left(\begin{array}{rr} 8369 & 8 \\ 8368 & 9 \end{array}\right),\left(\begin{array}{rr} 5588 & 1 \\ 2815 & 6 \end{array}\right),\left(\begin{array}{rr} 1051 & 1050 \\ 3154 & 7339 \end{array}\right)$.
The torsion field $K:=\Q(E[8376])$ is a degree-$22721823129600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8376\Z)$.
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 | $4$ | \( 3 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1745 = 5 \cdot 349 \) |
| $5$ | split multiplicative | $6$ | \( 2094 = 2 \cdot 3 \cdot 349 \) |
| $349$ | split multiplicative | $350$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 10470a
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{3}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{19 + \sqrt{349}})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1866240000.14 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 349 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | split | ord | ord | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | split |
| $\lambda$-invariant(s) | 1 | 1 | 2 | 3 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
| $\mu$-invariant(s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.