Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-14208x+529588\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-14208xz^2+529588z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-1150875x+389522250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-6, 784\right) \) | $0.34242353532461905943635528932$ | $\infty$ |
| \( \left(43, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([-6:784:1]\) | $0.34242353532461905943635528932$ | $\infty$ |
| \([43:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(-51, 21168\right) \) | $0.34242353532461905943635528932$ | $\infty$ |
| \( \left(390, 0\right) \) | $0$ | $2$ |
Integral points
\((-132,\pm 350)\), \((-38,\pm 1008)\), \((-6,\pm 784)\), \( \left(43, 0\right) \), \((92,\pm 98)\), \((218,\pm 2800)\), \((3179,\pm 179144)\)
\([-132:\pm 350:1]\), \([-38:\pm 1008:1]\), \([-6:\pm 784:1]\), \([43:0:1]\), \([92:\pm 98:1]\), \([218:\pm 2800:1]\), \([3179:\pm 179144:1]\)
\((-132,\pm 350)\), \((-38,\pm 1008)\), \((-6,\pm 784)\), \( \left(43, 0\right) \), \((92,\pm 98)\), \((218,\pm 2800)\), \((3179,\pm 179144)\)
Invariants
| Conductor: | $N$ | = | \( 2800 \) | = | $2^{4} \cdot 5^{2} \cdot 7$ |
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| Minimal Discriminant: | $\Delta$ | = | $60236288000000$ | = | $2^{15} \cdot 5^{6} \cdot 7^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{4956477625}{941192} \) | = | $2^{-3} \cdot 5^{3} \cdot 7^{-6} \cdot 11^{3} \cdot 31^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3616609728660251086007715244$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.13620516391097038811684026367$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0082122835525031$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.077038234059417$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.34242353532461905943635528932$ |
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| Real period: | $\Omega$ | ≈ | $0.59277770310210141078525885237$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ 2^{2}\cdot2\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.4357724410939476573450722698 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.435772441 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.592778 \cdot 0.342424 \cdot 48}{2^2} \\ & \approx 2.435772441\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6912 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | -1 | 4 | 15 | 3 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $6$ | $I_{6}$ | split 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 | 8.6.0.6 | $6$ |
| $3$ | 3Cs | 3.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1261 & 540 \\ 0 & 1541 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 2461 & 540 \\ 2190 & 781 \end{array}\right),\left(\begin{array}{rr} 2485 & 36 \\ 2484 & 37 \end{array}\right),\left(\begin{array}{rr} 2026 & 1515 \\ 2265 & 856 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 503 & 0 \\ 0 & 2519 \end{array}\right),\left(\begin{array}{rr} 179 & 1740 \\ 990 & 1349 \end{array}\right)$.
The torsion field $K:=\Q(E[2520])$ is a degree-$6688604160$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2520\Z)$.
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 | $4$ | \( 25 = 5^{2} \) |
| $3$ | good | $2$ | \( 400 = 2^{4} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 112 = 2^{4} \cdot 7 \) |
| $7$ | split multiplicative | $8$ | \( 400 = 2^{4} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 2800v
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14a2, its twist by $-20$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/6\Z\) | 2.2.60.1-98.1-l4 |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/6\Z\) | 2.0.20.1-98.2-a4 |
| $4$ | \(\Q(\sqrt{50 +10 \sqrt{-7}})\) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-5})\) | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{15})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.513802240000.7 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.6146560000.3 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | \(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{-5})\) | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.497871360000.82 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.6146560000.12 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/3\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.3458599644985044262957449216000000000.2 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.2745547906970040979968000000000.2 | \(\Z/18\Z\) | 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 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | ord | add | split | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 3 | - | 2 | 1,1 | 1 | 1 | 1 | 1,1 | 3 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 0 | - | 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.