Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-15573888x-2662042719\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-15573888xz^2-2662042719z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-20183758875x-123897508706250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(8571, 698273\right) \) | $5.8184446432109181132518790858$ | $\infty$ |
| \( \left(-\frac{685}{4}, \frac{681}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([8571:698273:1]\) | $5.8184446432109181132518790858$ | $\infty$ |
| \([-1370:681:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(308571, 151752744\right) \) | $5.8184446432109181132518790858$ | $\infty$ |
| \( \left(-6150, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(8571, 698273\right) \), \( \left(8571, -706845\right) \)
\([8571:698273:1]\), \([8571:-706845:1]\)
\((308571,\pm 151752744)\)
Invariants
| Conductor: | $N$ | = | \( 48450 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 17 \cdot 19$ |
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| Minimal Discriminant: | $\Delta$ | = | $238706130890310750000000$ | = | $2^{7} \cdot 3^{5} \cdot 5^{9} \cdot 17^{4} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{213887210383626155117}{122217539015839104} \) | = | $2^{-7} \cdot 3^{-5} \cdot 7^{3} \cdot 17^{-4} \cdot 19^{-6} \cdot 211^{3} \cdot 4049^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1755472590964932667387943938$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.9684688247709179857882248939$ |
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| $abc$ quality: | $Q$ | ≈ | $1.035728696799586$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.68180269954936$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.8184446432109181132518790858$ |
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| Real period: | $\Omega$ | ≈ | $0.082309992169865110027266755366$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 56 $ = $ 7\cdot1\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.7048258623287796920940101744 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.704825862 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.082310 \cdot 5.818445 \cdot 56}{2^2} \\ & \approx 6.704825862\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7526400 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $7$ | $I_{7}$ | split multiplicative | -1 | 1 | 7 | 7 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $17$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $19$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 |
|---|---|---|---|
| $2$ | 2B | 2.3.0.1 | $3$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 2277 & 4 \\ 2276 & 5 \end{array}\right),\left(\begin{array}{rr} 1828 & 1 \\ 1823 & 0 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 1139 & 0 \end{array}\right),\left(\begin{array}{rr} 1522 & 1 \\ 1519 & 0 \end{array}\right),\left(\begin{array}{rr} 1921 & 4 \\ 1562 & 9 \end{array}\right),\left(\begin{array}{rr} 1996 & 289 \\ 1425 & 856 \end{array}\right)$.
The torsion field $K:=\Q(E[2280])$ is a degree-$363095654400$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2280\Z)$.
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 | $4$ | \( 15 = 3 \cdot 5 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 850 = 2 \cdot 5^{2} \cdot 17 \) |
| $5$ | additive | $14$ | \( 646 = 2 \cdot 17 \cdot 19 \) |
| $7$ | good | $2$ | \( 24225 = 3 \cdot 5^{2} \cdot 17 \cdot 19 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 48450.y
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 48450.u1, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.4332000.3 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.10809345024000000.107 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.2.3698873646750000.7 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | ord | ord | ord | nonsplit | nonsplit | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 6 | 3 | - | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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.