Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2+37092x+4630205\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z+37092xz^2+4630205z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+48071205x+215305784694\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(1035, 33439\right) \) | $3.0763644502233132743956001705$ | $\infty$ |
| \( \left(-99, 49\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([1035:33439:1]\) | $3.0763644502233132743956001705$ | $\infty$ |
| \([-99:49:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(37275, 7334712\right) \) | $3.0763644502233132743956001705$ | $\infty$ |
| \( \left(-3549, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-99, 49\right) \), \( \left(1035, 33439\right) \), \( \left(1035, -34475\right) \)
\([-99:49:1]\), \([1035:33439:1]\), \([1035:-34475:1]\)
\( \left(-3549, 0\right) \), \((37275,\pm 7334712)\)
Invariants
| Conductor: | $N$ | = | \( 16562 \) | = | $2 \cdot 7^{2} \cdot 13^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-12465865820804032$ | = | $-1 \cdot 2^{6} \cdot 7^{9} \cdot 13^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{9938375}{21952} \) | = | $2^{-6} \cdot 5^{3} \cdot 7^{-3} \cdot 43^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7726509990674819777539637682$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.48277875419094304282545632430$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9869508090989833$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.551033564833338$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0763644502233132743956001705$ |
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| Real period: | $\Omega$ | ≈ | $0.27789847355337773084425066885$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 48 $ = $ ( 2 \cdot 3 )\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $10.259003817731218160035981475 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.259003818 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.277898 \cdot 3.076364 \cdot 48}{2^2} \\ & \approx 10.259003818\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 103680 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $7$ | $4$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $13$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 | 8.6.0.1 | $6$ |
| $3$ | 3Cs | 3.12.0.1 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6552 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 13 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 4535 & 0 \\ 0 & 6551 \end{array}\right),\left(\begin{array}{rr} 2341 & 780 \\ 3042 & 1171 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 3277 & 6084 \\ 0 & 2549 \end{array}\right),\left(\begin{array}{rr} 19 & 24 \\ 1440 & 1819 \end{array}\right),\left(\begin{array}{rr} 3277 & 6084 \\ 3042 & 4681 \end{array}\right),\left(\begin{array}{rr} 3938 & 6045 \\ 3627 & 6356 \end{array}\right),\left(\begin{array}{rr} 6517 & 36 \\ 6516 & 37 \end{array}\right)$.
The torsion field $K:=\Q(E[6552])$ is a degree-$365197787136$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6552\Z)$.
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 | $4$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $3$ | good | $2$ | \( 8281 = 7^{2} \cdot 13^{2} \) |
| $7$ | additive | $32$ | \( 338 = 2 \cdot 13^{2} \) |
| $13$ | additive | $86$ | \( 98 = 2 \cdot 7^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 16562.bv
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
The minimal quadratic twist of this elliptic curve is 14.a6, its twist by $-91$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{273}) \) | \(\Z/6\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-91}) \) | \(\Z/6\Z\) | 2.0.91.1-28.1-b3 |
| $4$ | 4.2.75712.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{-91})\) | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{-39})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-7}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.0.3440817243136.5 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.280883040256.10 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5554571841.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $8$ | 8.4.22751526260736.19 | \(\Z/12\Z\) | not in database |
| $8$ | 8.0.280883040256.14 | \(\Z/12\Z\) | 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 |
| $18$ | 18.6.1572514274247345012118995555813213769728.1 | \(\Z/18\Z\) | not in database |
| $18$ | 18.0.19504883286905868813156741826371.1 | \(\Z/18\Z\) | 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 | ord | ss | add | ss | add | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | 1 | 1,1 | - | 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 | 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.