Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-341501x-69817269\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-341501xz^2-69817269z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-442585323x-3256066746522\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2671/4, -2671/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 2310 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $444872222400201750$ | = | $2 \cdot 3^{4} \cdot 5^{3} \cdot 7 \cdot 11^{12} $ |
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| j-invariant: | $j$ | = | \( \frac{4404531606962679693649}{444872222400201750} \) | = | $2^{-1} \cdot 3^{-4} \cdot 5^{-3} \cdot 7^{-1} \cdot 11^{-12} \cdot 47^{3} \cdot 229^{3} \cdot 1523^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1224216312014297649249536570$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1224216312014297649249536570$ |
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| $abc$ quality: | $Q$ | ≈ | $1.013360225210077$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.434719539656582$ | |||
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: | $\Omega$ | ≈ | $0.19877877863678069949048305614$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 1\cdot2^{2}\cdot1\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.5780180154620525908286950105 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.578018015 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.198779 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 3.578018015\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 41472 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $5$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $11$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
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 | 4.6.0.1 |
| $3$ | 3B.1.2 | 3.8.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 4242 & 2719 \\ 7037 & 6692 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 7934 & 11 \end{array}\right),\left(\begin{array}{rr} 1336 & 3 \\ 6141 & 9154 \end{array}\right),\left(\begin{array}{rr} 2326 & 21 \\ 2025 & 8866 \end{array}\right),\left(\begin{array}{rr} 7701 & 6164 \\ 8 & 4653 \end{array}\right),\left(\begin{array}{rr} 2521 & 24 \\ 2532 & 289 \end{array}\right),\left(\begin{array}{rr} 9217 & 24 \\ 9216 & 25 \end{array}\right),\left(\begin{array}{rr} 3712 & 21 \\ 8955 & 8866 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$2452488192000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $4$ | \( 35 = 5 \cdot 7 \) |
| $3$ | split multiplicative | $4$ | \( 14 = 2 \cdot 7 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 462 = 2 \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | split multiplicative | $8$ | \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 2310t
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{70}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{7}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/6\Z\) | not in database |
| $3$ | 3.1.47628.2 | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{70})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{10})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{7})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.36294822144000.50 | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.254063755008.16 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.6805279152.4 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.254063755008000.6 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.7710244864000000.49 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.7965941760000.9 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.698450215471608010367748492705961687500000000.2 | \(\Z/18\Z\) | 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 |
|---|---|---|---|---|---|
| Reduction type | split | split | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | 3 | 3 | 0 | 3 | 0 |
| $\mu$-invariant(s) | 0 | 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 $0$.