Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2+2864x+5282264\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z+2864xz^2+5282264z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+231957x+3850074558\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-70, 2178)$ | $0.71378163852753728784876049104$ | $\infty$ |
| $(74, 2430)$ | $0.95972931019365071661197189824$ | $\infty$ |
| $(-169, 0)$ | $0$ | $2$ |
Integral points
\( \left(-169, 0\right) \), \((-166,\pm 510)\), \((-133,\pm 1602)\), \((-70,\pm 2178)\), \((74,\pm 2430)\), \((155,\pm 3078)\), \((194,\pm 3630)\), \((359,\pm 7260)\), \((722,\pm 19602)\), \((1019,\pm 32670)\), \((3802,\pm 234498)\), \((4394,\pm 291330)\), \((29234,\pm 4998510)\)
Invariants
| Conductor: | $N$ | = | \( 58080 \) | = | $2^{5} \cdot 3 \cdot 5 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $-12050945912332800$ | = | $-1 \cdot 2^{9} \cdot 3^{12} \cdot 5^{2} \cdot 11^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{2863288}{13286025} \) | = | $2^{3} \cdot 3^{-12} \cdot 5^{-2} \cdot 71^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7645880766168491809079365366$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.045780054797704926814040656523$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1738269518331081$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.05508035509916$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.66037988322465196513914729561$ |
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| Real period: | $\Omega$ | ≈ | $0.31567549547979212153308009856$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 192 $ = $ 2\cdot( 2^{2} \cdot 3 )\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $10.006355848407804983431033224 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 10.006355848 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.315675 \cdot 0.660380 \cdot 192}{2^2} \\ & \approx 10.006355848\end{aligned}$$
Modular invariants
Modular form 58080.2.a.be
For more coefficients, see the Downloads section to the right.
| Modular degree: | 491520 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_0^{*}$ | additive | -1 | 5 | 9 | 0 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $11$ | $4$ | $I_0^{*}$ | 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 | 8.12.0.16 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 264 = 2^{3} \cdot 3 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 89 & 176 \\ 44 & 177 \end{array}\right),\left(\begin{array}{rr} 119 & 0 \\ 0 & 263 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 258 & 259 \end{array}\right),\left(\begin{array}{rr} 257 & 8 \\ 256 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 144 & 209 \\ 187 & 78 \end{array}\right),\left(\begin{array}{rr} 144 & 209 \\ 121 & 12 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[264])$ is a degree-$20275200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/264\Z)$.
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 | $4$ | \( 121 = 11^{2} \) |
| $3$ | split multiplicative | $4$ | \( 19360 = 2^{5} \cdot 5 \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 11616 = 2^{5} \cdot 3 \cdot 11^{2} \) |
| $11$ | additive | $62$ | \( 480 = 2^{5} \cdot 3 \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 58080.be
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 480.a4, its twist by $44$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{22}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-11}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-11})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.153522012160000.23 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.4.4974113193984.59 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.194301296640000.66 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | nonsplit | ord | add | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 7 | 4 | 4 | - | 2 | 2 | 2 | 2,2 | 2 | 2 | 2 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.