Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2+12042x+2250119\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z+12042xz^2+2250119z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+15606000x+104794290000\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1557/4, 65021/8)$ | $1.2867192058847295112115845904$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 21675 \) | = | $3 \cdot 5^{2} \cdot 17^{2}$ |
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| Discriminant: | $\Delta$ | = | $-2291183307421875$ | = | $-1 \cdot 3^{5} \cdot 5^{8} \cdot 17^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{20480}{243} \) | = | $2^{12} \cdot 3^{-5} \cdot 5$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6282272941504854127399550443$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.86133798616702287711865182012$ |
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| $abc$ quality: | $Q$ | ≈ | $1.131038420174359$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.284147351201391$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2867192058847295112115845904$ |
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| Real period: | $\Omega$ | ≈ | $0.34016590103876227094676889024$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 30 $ = $ 5\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $13.130939941609790244404803878 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.130939942 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.340166 \cdot 1.286719 \cdot 30}{1^2} \\ & \approx 13.130939942\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 153600 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $5$ | $I_{5}$ | split multiplicative | -1 | 1 | 5 | 5 |
| $5$ | $3$ | $IV^{*}$ | additive | -1 | 2 | 8 | 0 |
| $17$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 501 & 10 \\ 500 & 11 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 455 & 391 \end{array}\right),\left(\begin{array}{rr} 269 & 0 \\ 0 & 509 \end{array}\right),\left(\begin{array}{rr} 509 & 170 \\ 0 & 203 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 341 & 340 \\ 85 & 171 \end{array}\right)$.
The torsion field $K:=\Q(E[510])$ is a degree-$225607680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/510\Z)$.
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 |
|---|---|---|---|
| $3$ | split multiplicative | $4$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 289 = 17^{2} \) |
| $17$ | additive | $146$ | \( 75 = 3 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 21675z
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 75c1, its twist by $85$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{85}) \) | \(\Z/5\Z\) | not in database |
| $3$ | 3.1.300.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.270000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.2210850000.1 | \(\Z/10\Z\) | not in database |
| $8$ | 8.2.9247184116875.2 | \(\Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | not in database |
| $20$ | 20.0.9387703148876316845417022705078125.2 | \(\Z/5\Z\) | 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 | ss | split | add | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 8,9 | 2 | - | 1 | 1 | 1 | - | 1 | 1 | 1 | 1 | 3 | 1 | 1 | 1 |
| $\mu$-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.