Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-107889505x+431373110497\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-107889505xz^2+431373110497z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8739049932x+314444780402544\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(6249, 34000)$ | $0$ | $4$ |
Integral points
\( \left(5999, 0\right) \), \((6249,\pm 34000)\)
Invariants
| Conductor: | $N$ | = | \( 16320 \) | = | $2^{6} \cdot 3 \cdot 5 \cdot 17$ |
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| Discriminant: | $\Delta$ | = | $20225376000000000000$ | = | $2^{17} \cdot 3^{7} \cdot 5^{12} \cdot 17^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{1059623036730633329075378}{154307373046875} \) | = | $2 \cdot 3^{-7} \cdot 5^{-12} \cdot 17^{-2} \cdot 179^{3} \cdot 251^{3} \cdot 1801^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1129365973321957390282386439$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.1309780915389398840204931385$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0554528133897931$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.917777647117481$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.16886625027717763076192215934$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot1\cdot( 2^{2} \cdot 3 )\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.0131975016630657845715329561 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 1.013197502 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.168866 \cdot 1.000000 \cdot 96}{4^2} \\ & \approx 1.013197502\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1720320 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{7}^{*}$ | additive | 1 | 6 | 17 | 0 |
| $3$ | $1$ | $I_{7}$ | nonsplit multiplicative | 1 | 1 | 7 | 7 |
| $5$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $17$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 | 4.12.0.7 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 120 = 2^{3} \cdot 3 \cdot 5 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 113 & 8 \\ 112 & 9 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 114 & 115 \end{array}\right),\left(\begin{array}{rr} 48 & 113 \\ 73 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 97 & 8 \\ 28 & 33 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 44 & 1 \\ 103 & 6 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 104 & 37 \\ 71 & 42 \end{array}\right)$.
The torsion field $K:=\Q(E[120])$ is a degree-$737280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/120\Z)$.
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 | $4$ | \( 3 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 1088 = 2^{6} \cdot 17 \) |
| $5$ | split multiplicative | $6$ | \( 3264 = 2^{6} \cdot 3 \cdot 17 \) |
| $7$ | good | $2$ | \( 5440 = 2^{6} \cdot 5 \cdot 17 \) |
| $17$ | nonsplit multiplicative | $18$ | \( 960 = 2^{6} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 16320.v
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2040.h1, its twist by $8$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.4.44390400.2 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.1021511146143744.22 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.1866240000.5 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 17 |
|---|---|---|---|---|
| Reduction type | add | nonsplit | split | nonsplit |
| $\lambda$-invariant(s) | - | 2 | 1 | 0 |
| $\mu$-invariant(s) | - | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.