Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+164020x+597488397\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+164020xz^2+597488397z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+2624325x+38241881750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(30413/16, 5582169/64)$ | $8.4165134449917019956926816027$ | $\infty$ |
| $(-3109/4, 3105/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 3150 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $-154524284362792968750$ | = | $-1 \cdot 2 \cdot 3^{10} \cdot 5^{18} \cdot 7^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{42841933504271}{13565917968750} \) | = | $2^{-1} \cdot 3^{-4} \cdot 5^{-12} \cdot 7^{-3} \cdot 11^{3} \cdot 3181^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.5530511135870297918239448858$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1990260130359247588259426007$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0876511700840974$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.696302596725$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $8.4165134449917019956926816027$ |
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| Real period: | $\Omega$ | ≈ | $0.14149108174105607774218063473$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.7634463672800735728411765322 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.763446367 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.141491 \cdot 8.416513 \cdot 16}{2^2} \\ & \approx 4.763446367\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 110592 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $4$ | $I_{12}^{*}$ | additive | 1 | 2 | 18 | 12 |
| $7$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 | 8.12.0.6 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 246 & 409 \\ 35 & 176 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 699 & 556 \\ 832 & 387 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 374 & 11 \end{array}\right),\left(\begin{array}{rr} 631 & 24 \\ 642 & 289 \end{array}\right),\left(\begin{array}{rr} 503 & 816 \\ 156 & 551 \end{array}\right),\left(\begin{array}{rr} 817 & 24 \\ 816 & 25 \end{array}\right),\left(\begin{array}{rr} 496 & 21 \\ 195 & 466 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[840])$ is a degree-$185794560$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/840\Z)$.
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 | $4$ | \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $8$ | \( 50 = 2 \cdot 5^{2} \) |
| $5$ | additive | $18$ | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 3150bf
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210a8, its twist by $-15$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-14}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-105}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-14}, \sqrt{30})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.3225600.7 | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-14})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{6})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{-21})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.354294000.5 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.24395696640000.26 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.2039281090560000.34 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.7965941760000.27 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.4.10404495360000.5 | \(\Z/24\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | 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 | \(\Z/16\Z\) | 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 | \(\Z/2\Z \oplus \Z/24\Z\) | not in database |
| $18$ | 18.6.22264049152039574408495695312500000000.3 | \(\Z/18\Z\) | 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 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | split | add | add | nonsplit | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2 | - | - | 1 | 1,1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 2 | - | - | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.