Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-578x+2756\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-578xz^2+2756z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-748467x+130840974\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z \oplus \Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(5, -3\right) \) | $0$ | $2$ |
| \( \left(0, 52\right) \) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([5:-3:1]\) | $0$ | $2$ |
| \([0:52:1]\) | $0$ | $6$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(183, 0\right) \) | $0$ | $2$ |
| \( \left(3, 11340\right) \) | $0$ | $6$ |
Integral points
\( \left(-15, 97\right) \), \( \left(-15, -83\right) \), \( \left(0, 52\right) \), \( \left(0, -53\right) \), \( \left(5, -3\right) \), \( \left(21, -11\right) \), \( \left(30, 97\right) \), \( \left(30, -128\right) \), \( \left(105, 997\right) \), \( \left(105, -1103\right) \)
\([-15:97:1]\), \([-15:-83:1]\), \([0:52:1]\), \([0:-53:1]\), \([5:-3:1]\), \([21:-11:1]\), \([30:97:1]\), \([30:-128:1]\), \([105:997:1]\), \([105:-1103:1]\)
\((-537,\pm 19440)\), \((3,\pm 11340)\), \( \left(183, 0\right) \), \( \left(759, 0\right) \), \((1083,\pm 24300)\), \((3783,\pm 226800)\)
Invariants
| Conductor: | $N$ | = | \( 210 \) | = | $2 \cdot 3 \cdot 5 \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $8930250000$ | = | $2^{4} \cdot 3^{6} \cdot 5^{6} \cdot 7^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{21302308926361}{8930250000} \) | = | $2^{-4} \cdot 3^{-6} \cdot 5^{-6} \cdot 7^{-2} \cdot 19^{3} \cdot 1459^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.60638527169642118027051752577$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.60638527169642118027051752577$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0136164980344768$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.739521116185548$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $1.1766630746428014303624577180$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 144 $ = $ 2\cdot( 2 \cdot 3 )\cdot( 2 \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $12$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.1766630746428014303624577180 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.176663075 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.176663 \cdot 1.000000 \cdot 144}{12^2} \\ & \approx 1.176663075\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 192 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $5$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $2$ | $I_{2}$ | 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 |
|---|---|---|---|
| $2$ | 2Cs | 4.12.0.1 | $12$ |
| $3$ | 3B.1.1 | 3.8.0.1 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 409 & 12 \\ 408 & 13 \end{array}\right),\left(\begin{array}{rr} 147 & 10 \\ 142 & 3 \end{array}\right),\left(\begin{array}{rr} 9 & 4 \\ 404 & 413 \end{array}\right),\left(\begin{array}{rr} 337 & 12 \\ 342 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 234 & 415 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 211 & 12 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[420])$ is a degree-$11612160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/420\Z)$.
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 | $4$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 14 = 2 \cdot 7 \) |
| $5$ | split multiplicative | $6$ | \( 42 = 2 \cdot 3 \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 30 = 2 \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 210b
consists of 8 curves linked by isogenies of
degrees dividing 12.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{21}) \) | \(\Z/2\Z \oplus \Z/12\Z\) | 2.2.21.1-2100.1-n5 |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-21})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{5})\) | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $6$ | 6.0.1037232.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.31116960000.8 | \(\Z/4\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.4.8782450790400.27 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $9$ | 9.3.8612931426750000.6 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $12$ | 12.0.52716660869376.1 | \(\Z/6\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.6.33663242317883836505645491312500000000.1 | \(\Z/2\Z \oplus \Z/36\Z\) | 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 |
|---|---|---|---|---|
| Reduction type | nonsplit | split | split | split |
| $\lambda$-invariant(s) | 0 | 1 | 3 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.