Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-484203x-123204202\)
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(homogenize, simplify) |
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\(y^2z=x^3-484203xz^2-123204202z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-484203x-123204202\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10174, 1023750\right) \) | $4.2937856061255936515509240134$ | $\infty$ |
| \( \left(-326, 0\right) \) | $0$ | $2$ |
| \( \left(799, 0\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([10174:1023750:1]\) | $4.2937856061255936515509240134$ | $\infty$ |
| \([-326:0:1]\) | $0$ | $2$ |
| \([799:0:1]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(10174, 1023750\right) \) | $4.2937856061255936515509240134$ | $\infty$ |
| \( \left(-326, 0\right) \) | $0$ | $2$ |
| \( \left(799, 0\right) \) | $0$ | $2$ |
Integral points
\( \left(-473, 0\right) \), \( \left(-326, 0\right) \), \( \left(799, 0\right) \), \((10174,\pm 1023750)\)
\([-473:0:1]\), \([-326:0:1]\), \([799:0:1]\), \([10174:\pm 1023750:1]\)
\( \left(-473, 0\right) \), \( \left(-326, 0\right) \), \( \left(799, 0\right) \), \((10174,\pm 1023750)\)
Invariants
| Conductor: | $N$ | = | \( 133560 \) | = | $2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 53$ |
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| Minimal Discriminant: | $\Delta$ | = | $708001079184000000$ | = | $2^{10} \cdot 3^{8} \cdot 5^{6} \cdot 7^{4} \cdot 53^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{16818166777456804}{948432515625} \) | = | $2^{2} \cdot 3^{-2} \cdot 5^{-6} \cdot 7^{-4} \cdot 53^{-2} \cdot 103^{3} \cdot 1567^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.1798147491467673142248334392$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0528859543460913773461840529$ |
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| $abc$ quality: | $Q$ | ≈ | $0.916669883599156$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.311393145116363$ | |||
| 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: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.2937856061255936515509240134$ |
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| Real period: | $\Omega$ | ≈ | $0.18162165467791337203496578265$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 128 $ = $ 2\cdot2^{2}\cdot2\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $6.2387555729339002381630328531 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.238755573 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.181622 \cdot 4.293786 \cdot 128}{4^2} \\ & \approx 6.238755573\end{aligned}$$
Modular invariants
Modular form 133560.2.a.q
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1474560 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $III^{*}$ | additive | 1 | 3 | 10 | 0 |
| $3$ | $4$ | $I_{2}^{*}$ | additive | -1 | 2 | 8 | 2 |
| $5$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $53$ | $2$ | $I_{2}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cs | 2.6.0.1 | $6$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 6360 = 2^{3} \cdot 3 \cdot 5 \cdot 53 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 3181 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 1591 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3817 & 4 \\ 1274 & 9 \end{array}\right),\left(\begin{array}{rr} 6357 & 4 \\ 6356 & 5 \end{array}\right),\left(\begin{array}{rr} 2119 & 6356 \\ 4238 & 6351 \end{array}\right),\left(\begin{array}{rr} 2281 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[6360])$ is a degree-$5705697853440$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6360\Z)$.
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 | $2$ | \( 9 = 3^{2} \) |
| $3$ | additive | $8$ | \( 2968 = 2^{3} \cdot 7 \cdot 53 \) |
| $5$ | nonsplit multiplicative | $6$ | \( 26712 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 53 \) |
| $7$ | split multiplicative | $8$ | \( 19080 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 53 \) |
| $53$ | nonsplit multiplicative | $54$ | \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 133560bi
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 44520s2, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{-3}, \sqrt{106})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{318})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{3}, \sqrt{-5})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | 53 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | add | nonsplit | split | ss | ord | ord | ord | ss | ord | ord | ord | ord | ss | ss | nonsplit |
| $\lambda$-invariant(s) | - | - | 1 | 2 | 1,1 | 3 | 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 | 0,0 | 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.