Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-3179x+76593\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-3179xz^2+76593z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4120011x+3585883014\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(22, 121\right) \) | $0.068249777583275577026477431061$ | $\infty$ |
| \( \left(-66, 33\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([22:121:1]\) | $0.068249777583275577026477431061$ | $\infty$ |
| \([-66:33:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(795, 28512\right) \) | $0.068249777583275577026477431061$ | $\infty$ |
| \( \left(-2373, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-66, 33\right) \), \( \left(-44, 385\right) \), \( \left(-44, -341\right) \), \( \left(-38, 397\right) \), \( \left(-38, -359\right) \), \( \left(-2, 289\right) \), \( \left(-2, -287\right) \), \( \left(22, 121\right) \), \( \left(22, -143\right) \), \( \left(34, 73\right) \), \( \left(34, -107\right) \), \( \left(46, 145\right) \), \( \left(46, -191\right) \), \( \left(78, 513\right) \), \( \left(78, -591\right) \), \( \left(88, 649\right) \), \( \left(88, -737\right) \), \( \left(286, 4609\right) \), \( \left(286, -4895\right) \), \( \left(382, 7201\right) \), \( \left(382, -7583\right) \), \( \left(15334, 1891153\right) \), \( \left(15334, -1906487\right) \)
\([-66:33:1]\), \([-44:385:1]\), \([-44:-341:1]\), \([-38:397:1]\), \([-38:-359:1]\), \([-2:289:1]\), \([-2:-287:1]\), \([22:121:1]\), \([22:-143:1]\), \([34:73:1]\), \([34:-107:1]\), \([46:145:1]\), \([46:-191:1]\), \([78:513:1]\), \([78:-591:1]\), \([88:649:1]\), \([88:-737:1]\), \([286:4609:1]\), \([286:-4895:1]\), \([382:7201:1]\), \([382:-7583:1]\), \([15334:1891153:1]\), \([15334:-1906487:1]\)
\( \left(-2373, 0\right) \), \((-1581,\pm 78408)\), \((-1365,\pm 81648)\), \((-69,\pm 62208)\), \((795,\pm 28512)\), \((1227,\pm 19440)\), \((1659,\pm 36288)\), \((2811,\pm 119232)\), \((3171,\pm 149688)\), \((10299,\pm 1026432)\), \((13755,\pm 1596672)\), \((552027,\pm 410145120)\)
Invariants
| Conductor: | $N$ | = | \( 3234 \) | = | $2 \cdot 3 \cdot 7^{2} \cdot 11$ |
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| Minimal Discriminant: | $\Delta$ | = | $-495709175808$ | = | $-1 \cdot 2^{14} \cdot 3^{6} \cdot 7^{3} \cdot 11^{2} $ |
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| j-invariant: | $j$ | = | \( -\frac{10358806345399}{1445216256} \) | = | $-1 \cdot 2^{-14} \cdot 3^{-6} \cdot 11^{-2} \cdot 21799^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.97555181954379017163368591809$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.48907428227996184535734773223$ |
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| $abc$ quality: | $Q$ | ≈ | $0.998022530489296$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.457425520118723$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.068249777583275577026477431061$ |
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| Real period: | $\Omega$ | ≈ | $0.90101396661302936623310839317$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 336 $ = $ ( 2 \cdot 7 )\cdot( 2 \cdot 3 )\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.1654962369441878292992399969 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.165496237 \approx L'(E,1) & = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.901014 \cdot 0.068250 \cdot 336}{2^2} \\ & \approx 5.165496237\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5376 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $14$ | $I_{14}$ | split multiplicative | -1 | 1 | 14 | 14 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
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 | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 616 = 2^{3} \cdot 7 \cdot 11 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 613 & 4 \\ 612 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 309 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 57 & 4 \\ 114 & 9 \end{array}\right),\left(\begin{array}{rr} 180 & 1 \\ 263 & 0 \end{array}\right),\left(\begin{array}{rr} 233 & 386 \\ 384 & 231 \end{array}\right)$.
The torsion field $K:=\Q(E[616])$ is a degree-$3406233600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/616\Z)$.
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 | $4$ | \( 7 \) |
| $3$ | split multiplicative | $4$ | \( 1078 = 2 \cdot 7^{2} \cdot 11 \) |
| $7$ | additive | $20$ | \( 33 = 3 \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 3234v
consists of 2 curves linked by isogenies of
degree 2.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{49 +4 \sqrt{154}})\) | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.1180751717376.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.7055355940864.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 |
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 | split | ord | add | split | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 2 | 1 | - | 2 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.