Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-946542x+263932605\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-946542xz^2+263932605z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15144672x+16891686736\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(833, 7312\right) \) | $2.5864440063208352335100497097$ | $\infty$ |
| \( \left(1083, 22562\right) \) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([833:7312:1]\) | $2.5864440063208352335100497097$ | $\infty$ |
| \([1083:22562:1]\) | $0$ | $3$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3332, 58500\right) \) | $2.5864440063208352335100497097$ | $\infty$ |
| \( \left(4332, 180500\right) \) | $0$ | $3$ |
Integral points
\( \left(833, 7312\right) \), \( \left(833, -7313\right) \), \( \left(1083, 22562\right) \), \( \left(1083, -22563\right) \), \( \left(1805, 66604\right) \), \( \left(1805, -66605\right) \)
\([833:7312:1]\), \([833:-7313:1]\), \([1083:22562:1]\), \([1083:-22563:1]\), \([1805:66604:1]\), \([1805:-66605:1]\)
\((3332,\pm 58500)\), \((4332,\pm 180500)\), \((7220,\pm 532836)\)
Invariants
| Conductor: | $N$ | = | \( 16245 \) | = | $3^{2} \cdot 5 \cdot 19^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $24181674720486328125$ | = | $3^{6} \cdot 5^{9} \cdot 19^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{7575076864}{1953125} \) | = | $2^{15} \cdot 5^{-9} \cdot 19 \cdot 23^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4286059809917377725314143025$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.083659482786610713172226603886$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0058640564899433$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.455633461984215$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.5864440063208352335100497097$ |
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| Real period: | $\Omega$ | ≈ | $0.19931237549829245326802653670$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 54 $ = $ 2\cdot3^{2}\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.0930617939587572691473583015 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.093061794 \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 0.199312 \cdot 2.586444 \cdot 54}{3^2} \\ & \approx 3.093061794\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 344736 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $19$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|---|
| $3$ | 3B.1.1 | 9.24.0.2 | $24$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1710 = 2 \cdot 3^{2} \cdot 5 \cdot 19 \), index $144$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 12 & 217 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 1693 & 18 \\ 1692 & 19 \end{array}\right),\left(\begin{array}{rr} 85 & 1692 \\ 1152 & 1321 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 1017 & 1702 \end{array}\right),\left(\begin{array}{rr} 1709 & 1692 \\ 0 & 1519 \end{array}\right),\left(\begin{array}{rr} 13 & 12 \\ 1646 & 1651 \end{array}\right)$.
The torsion field $K:=\Q(E[1710])$ is a degree-$9573811200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1710\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $2$ | \( 361 = 19^{2} \) |
| $5$ | split multiplicative | $6$ | \( 3249 = 3^{2} \cdot 19^{2} \) |
| $19$ | additive | $146$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 16245e
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 1805b2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.3.7220.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.6.260642000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.3518667.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.75006330133563.4 | \(\Z/9\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.16877848680315122776257224907.2 | \(\Z/3\Z \oplus \Z/9\Z\) | not in database |
| $18$ | 18.0.2788139363326671805632000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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 | ss | add | split | ord | ord | ord | ord | add | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2,11 | - | 4 | 1 | 1 | 3 | 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.