Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+7225x+179885\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+7225xz^2+179885z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+9363573x+8252269254\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(33, 658)$ | $0.45038779947725862527332846368$ | $\infty$ |
| $(-93/4, 89/8)$ | $0$ | $2$ |
Integral points
\( \left(-11, 320\right) \), \( \left(-11, -310\right) \), \( \left(33, 658\right) \), \( \left(33, -692\right) \), \( \left(133, 1808\right) \), \( \left(133, -1942\right) \), \( \left(483, 10558\right) \), \( \left(483, -11042\right) \)
Invariants
| Conductor: | $N$ | = | \( 3570 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $-37652343750000$ | = | $-1 \cdot 2^{4} \cdot 3^{4} \cdot 5^{12} \cdot 7 \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{41709358422320399}{37652343750000} \) | = | $2^{-4} \cdot 3^{-4} \cdot 5^{-12} \cdot 7^{-1} \cdot 17^{-1} \cdot 241^{3} \cdot 1439^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2923090134834986067690678194$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2923090134834986067690678194$ |
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| $abc$ quality: | $Q$ | ≈ | $0.973167105198145$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.678239705626807$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.45038779947725862527332846368$ |
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| Real period: | $\Omega$ | ≈ | $0.42356548912851885916079360709$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot2\cdot( 2^{2} \cdot 3 )\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.5784494859944556838909443971 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.578449486 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.423565 \cdot 0.450388 \cdot 96}{2^2} \\ & \approx 4.578449486\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 12288 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $3$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $5$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 4.12.0.8 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14280 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 17 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 14273 & 8 \\ 14272 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5363 & 5356 \\ 5402 & 12501 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 6128 & 3 \\ 2045 & 2 \end{array}\right),\left(\begin{array}{rr} 2857 & 8 \\ 11428 & 33 \end{array}\right),\left(\begin{array}{rr} 9521 & 8 \\ 9524 & 33 \end{array}\right),\left(\begin{array}{rr} 12499 & 12498 \\ 8938 & 1795 \end{array}\right),\left(\begin{array}{rr} 10088 & 3 \\ 13445 & 2 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 14274 & 14275 \end{array}\right)$.
The torsion field $K:=\Q(E[14280])$ is a degree-$116435221217280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14280\Z)$.
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 | $4$ | \( 119 = 7 \cdot 17 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 238 = 2 \cdot 7 \cdot 17 \) |
| $5$ | split multiplicative | $6$ | \( 714 = 2 \cdot 3 \cdot 7 \cdot 17 \) |
| $7$ | split multiplicative | $8$ | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 3570u
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-119}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{119}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{119})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.0.6740636.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.726978778951936.1 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.2.568383720054192.2 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | nonsplit | split | split | ord | ord | split | ss | ord | ord | ss | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | 1 | 2 | 2 | 1 | 1 | 2 | 1,1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0,0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.