Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-1664935x+826189157\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-1664935xz^2+826189157z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2157755787x+38579047654086\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(745, -354)$ | $3.0237738248942891719276314428$ | $\infty$ |
| $(2979/4, -2983/8)$ | $0$ | $2$ |
Integral points
\( \left(745, -354\right) \), \( \left(745, -392\right) \), \( \left(4285, 266916\right) \), \( \left(4285, -271202\right) \)
Invariants
| Conductor: | $N$ | = | \( 60690 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $62590647422520$ | = | $2^{3} \cdot 3^{3} \cdot 5 \cdot 7^{4} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{21145699168383889}{2593080} \) | = | $2^{-3} \cdot 3^{-3} \cdot 5^{-1} \cdot 7^{-4} \cdot 11^{3} \cdot 23^{3} \cdot 1093^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.0663265407299779532530872912$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.64971986870186991312831998226$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0250305664622372$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.956582683669109$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.0237738248942891719276314428$ |
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| Real period: | $\Omega$ | ≈ | $0.48274893759418883783219187217$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 3\cdot1\cdot1\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.7583416089570093400000819079 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.758341609 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.482749 \cdot 3.023774 \cdot 24}{2^2} \\ & \approx 8.758341609\end{aligned}$$
Modular invariants
Modular form 60690.2.a.bp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 884736 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $17$ | $4$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
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 $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 5879 & 0 \\ 0 & 14279 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 5254 & 1683 \\ 255 & 4234 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 12974 & 5051 \end{array}\right),\left(\begin{array}{rr} 6121 & 13464 \\ 8772 & 4489 \end{array}\right),\left(\begin{array}{rr} 14257 & 24 \\ 14256 & 25 \end{array}\right),\left(\begin{array}{rr} 8416 & 4641 \\ 9095 & 13346 \end{array}\right),\left(\begin{array}{rr} 13686 & 10489 \\ 10115 & 7736 \end{array}\right),\left(\begin{array}{rr} 11272 & 11781 \\ 11475 & 3826 \end{array}\right)$.
The torsion field $K:=\Q(E[14280])$ is a degree-$14554402652160$ 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 | $4$ | \( 4335 = 3 \cdot 5 \cdot 17^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 10115 = 5 \cdot 7 \cdot 17^{2} \) |
| $5$ | split multiplicative | $6$ | \( 12138 = 2 \cdot 3 \cdot 7 \cdot 17^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \) |
| $17$ | additive | $146$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 60690.bp
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 210.d2, its twist by $17$.
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 |
| $2$ | \(\Q(\sqrt{255}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{34}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{34})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{30})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{17})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{2}, \sqrt{17})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.199059406875.6 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | 8.8.277102632960000.1 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | \(\Z/24\Z\) | not in database |
| $18$ | 18.6.225207182226836450197900578718324800000000.2 | \(\Z/18\Z\) | 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 | split | nonsplit | split | nonsplit | ss | ord | add | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 10 | 1 | 2 | 1 | 1,1 | 1 | - | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3 |
| $\mu$-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.