Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-28467x-1810809\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-28467xz^2-1810809z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-455475x-116347250\)
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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 |
|---|---|---|
| $(-87, 3)$ | $2.1343413720492027079046483605$ | $\infty$ |
| $(-95, 219)$ | $2.3995861154191094498653265183$ | $\infty$ |
| $(-429/4, 429/8)$ | $0$ | $2$ |
Integral points
\( \left(-101, 213\right) \), \( \left(-101, -112\right) \), \( \left(-95, 219\right) \), \( \left(-95, -124\right) \), \( \left(-87, 84\right) \), \( \left(-87, 3\right) \), \( \left(199, 513\right) \), \( \left(199, -712\right) \), \( \left(399, 6888\right) \), \( \left(399, -7287\right) \), \( \left(885, 25356\right) \), \( \left(885, -26241\right) \), \( \left(5005, 351351\right) \), \( \left(5005, -356356\right) \)
Invariants
| Conductor: | $N$ | = | \( 34650 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $48735723093750$ | = | $2 \cdot 3^{10} \cdot 5^{6} \cdot 7^{4} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( \frac{223980311017}{4278582} \) | = | $2^{-1} \cdot 3^{-4} \cdot 7^{-4} \cdot 11^{-1} \cdot 6073^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.4196856959798947175861366818$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.065660595428789684588134396727$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9368058527651448$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.05461688327626$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.3226490588539997571193975962$ |
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| Real period: | $\Omega$ | ≈ | $0.36797415049948809150595365077$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 1\cdot2^{2}\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $6.3624924613568490535328313508 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 6.362492461 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.367974 \cdot 4.322649 \cdot 16}{2^2} \\ & \approx 6.362492461\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 131072 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $3$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $11$ | $1$ | $I_{1}$ | nonsplit 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 | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 5543 & 0 \\ 0 & 9239 \end{array}\right),\left(\begin{array}{rr} 5776 & 8555 \\ 235 & 7656 \end{array}\right),\left(\begin{array}{rr} 9233 & 8 \\ 9232 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 3536 & 7395 \\ 3365 & 5546 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 696 & 1625 \\ 2995 & 8526 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 9234 & 9235 \end{array}\right),\left(\begin{array}{rr} 3079 & 1840 \\ 8620 & 7359 \end{array}\right),\left(\begin{array}{rr} 5281 & 7400 \\ 6340 & 1881 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$19619905536000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $4$ | \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \) |
| $3$ | additive | $8$ | \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \) |
| $5$ | additive | $14$ | \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 34650.a
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 462.a2, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{22}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-165}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-30}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{22}, \sqrt{-30})\) | \(\Z/2\Z \oplus \Z/4\Z\) | 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 |
| $8$ | 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 | nonsplit | add | add | nonsplit | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord |
| $\lambda$-invariant(s) | 5 | - | - | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 2,2 | 2 |
| $\mu$-invariant(s) | 1 | - | - | 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.