Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-374124865x+2848350713375\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-374124865xz^2+2848350713375z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-30304114092x+2076538582392624\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-22391, 0)$ | $0$ | $2$ |
Integral points
\( \left(-22391, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 47040 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 7^{2}$ |
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| Discriminant: | $\Delta$ | = | $-153740419343960822867558400$ | = | $-1 \cdot 2^{19} \cdot 3 \cdot 5^{2} \cdot 7^{22} $ |
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| j-invariant: | $j$ | = | \( -\frac{187778242790732059201}{4984939585440150} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-1} \cdot 5^{-2} \cdot 7^{-16} \cdot 241^{3} \cdot 23761^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.8071052622631376636315628013$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7944294168955630469530382474$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0554678196515268$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.588008085531551$ | |||
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: | $\Omega$ | ≈ | $0.057581818672342353255199044297$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $3.6852363950299106083327388350 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $16$ = $4^2$ (exact) |
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BSD formula
$$\begin{aligned} 3.685236395 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{16 \cdot 0.057582 \cdot 1.000000 \cdot 16}{2^2} \\ & \approx 3.685236395\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18874368 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{9}^{*}$ | additive | 1 | 6 | 19 | 1 |
| $3$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
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 | 32.96.0.58 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1244 & 3331 \\ 2689 & 2590 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1439 & 3328 \\ 2864 & 2847 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 30 & 31 \\ 2089 & 3204 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 2711 & 26 \\ 2118 & 875 \end{array}\right),\left(\begin{array}{rr} 1471 & 32 \\ 2326 & 513 \end{array}\right),\left(\begin{array}{rr} 3329 & 32 \\ 3328 & 33 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 798 & 1355 \end{array}\right)$.
The torsion field $K:=\Q(E[3360])$ is a degree-$23781703680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3360\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 147 = 3 \cdot 7^{2} \) |
| $3$ | split multiplicative | $4$ | \( 15680 = 2^{6} \cdot 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \) |
| $7$ | additive | $32$ | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 47040.ha
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 210.e3, its twist by $-56$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-6}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{14}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-21}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-6}, \sqrt{14})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{7})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{14})\) | \(\Z/8\Z\) | not in database |
| $8$ | 8.8.25176309760000.1 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.4588382453760000.286 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.18353529815040000.43 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.12745506816.4 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.2039281090560000.357 | \(\Z/16\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | 16.0.4158667366315582921113600000000.10 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/32\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 |
|---|---|---|---|---|
| Reduction type | add | split | split | add |
| $\lambda$-invariant(s) | - | 5 | 1 | - |
| $\mu$-invariant(s) | - | 0 | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.