Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-937917x+69004741\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-937917xz^2+69004741z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-15006675x+4401296750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-2953/4, 154459/8)$ | $7.4822242890440551691464205607$ | $\infty$ |
| $(74, -37)$ | $0$ | $2$ |
Integral points
\( \left(74, -37\right) \)
Invariants
| Conductor: | $N$ | = | \( 7650 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $50761728000000000000$ | = | $2^{24} \cdot 3^{6} \cdot 5^{12} \cdot 17 $ |
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| j-invariant: | $j$ | = | \( \frac{8010684753304969}{4456448000000} \) | = | $2^{-24} \cdot 5^{-6} \cdot 17^{-1} \cdot 19^{3} \cdot 10531^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.4715015775931382283630625019$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.1174764770420331953650602168$ |
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| $abc$ quality: | $Q$ | ≈ | $1.042560452096738$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.912002778529456$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.4822242890440551691464205607$ |
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| Real period: | $\Omega$ | ≈ | $0.17340815548806447457346630619$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.5949574258222483737460159767 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.594957426 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.173408 \cdot 7.482224 \cdot 8}{2^2} \\ & \approx 2.594957426\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 276480 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{24}$ | nonsplit multiplicative | 1 | 1 | 24 | 24 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $I_{6}^{*}$ | additive | 1 | 2 | 12 | 6 |
| $17$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
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 | 4.6.0.3 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 2025 & 1934 \\ 82 & 1069 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 21 & 4 \\ 1940 & 2021 \end{array}\right),\left(\begin{array}{rr} 511 & 24 \\ 12 & 289 \end{array}\right),\left(\begin{array}{rr} 2017 & 24 \\ 2016 & 25 \end{array}\right),\left(\begin{array}{rr} 7 & 24 \\ 492 & 1687 \end{array}\right),\left(\begin{array}{rr} 1021 & 24 \\ 1032 & 289 \end{array}\right),\left(\begin{array}{rr} 258 & 1 \\ 1823 & 8 \end{array}\right),\left(\begin{array}{rr} 679 & 2016 \\ 680 & 2039 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$7219445760$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 3825 = 3^{2} \cdot 5^{2} \cdot 17 \) |
| $3$ | additive | $2$ | \( 425 = 5^{2} \cdot 17 \) |
| $5$ | additive | $18$ | \( 306 = 2 \cdot 3^{2} \cdot 17 \) |
| $17$ | split multiplicative | $18$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 7650y
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 170b3, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.61200.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{17})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.281883375.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.1082432160000.15 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.19551430890000.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.3745440000.9 | \(\Z/12\Z\) | 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 |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | 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 |
| $18$ | 18.6.2216429933248706293261125000000000000.2 | \(\Z/18\Z\) | 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 | nonsplit | add | add | ord | ord | ord | split | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | - | 1 | 3 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\mu$-invariant(s) | 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.