Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2+28048x+1228466\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z+28048xz^2+1228466z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+448773x+79070614\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{23551}{4}, \frac{3591903}{8}\right) \) | $10.081415734616332598215760116$ | $\infty$ |
| \( \left(-\frac{165}{4}, \frac{161}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([47102:3591903:8]\) | $10.081415734616332598215760116$ | $\infty$ |
| \([-330:161:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(23550, 3615458\right) \) | $10.081415734616332598215760116$ | $\infty$ |
| \( \left(-166, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 7335 \) | = | $3^{2} \cdot 5 \cdot 163$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2071620650088135$ | = | $-1 \cdot 3^{26} \cdot 5 \cdot 163 $ |
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| j-invariant: | $j$ | = | \( \frac{3347467708032071}{2841729286815} \) | = | $3^{-20} \cdot 5^{-1} \cdot 13^{3} \cdot 37^{3} \cdot 163^{-1} \cdot 311^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.6265521046210509560682003526$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.0772459602869961103705777341$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0048043387596701$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.756931588541711$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $2$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $10.081415734616332598215760116$ |
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| Real period: | $\Omega$ | ≈ | $0.30139420430321794585540938320$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.0384802735846309792032581895 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.038480274 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.301394 \cdot 10.081416 \cdot 4}{2^2} \\ & \approx 3.038480274\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 30720 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $4$ | $I_{20}^{*}$ | additive | -1 | 2 | 26 | 20 |
| $5$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $163$ | $1$ | $I_{1}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.12.0.6 | $12$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19560 = 2^{3} \cdot 3 \cdot 5 \cdot 163 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 19553 & 8 \\ 19552 & 9 \end{array}\right),\left(\begin{array}{rr} 13039 & 19552 \\ 13036 & 19527 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7339 & 7336 \\ 17138 & 7341 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 15656 & 3 \\ 5 & 2 \end{array}\right),\left(\begin{array}{rr} 15004 & 1 \\ 3143 & 6 \end{array}\right),\left(\begin{array}{rr} 7339 & 7338 \\ 2458 & 12235 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 19554 & 19555 \end{array}\right)$.
The torsion field $K:=\Q(E[19560])$ is a degree-$517242181386240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19560\Z)$.
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 |
|---|---|---|---|
| $3$ | additive | $8$ | \( 815 = 5 \cdot 163 \) |
| $5$ | split multiplicative | $6$ | \( 1467 = 3^{2} \cdot 163 \) |
| $163$ | nonsplit multiplicative | $164$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 7335c
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2445c4, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-815}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-489}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{15}, \sqrt{-489})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.46944000.1 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.78156344203666875.7 | \(\Z/6\Z\) | 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 | \(\Z/16\Z\) | 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 | 163 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | ord | add | split | ss | ord | ord | ord | ss | ord | ord | ord | ord | ord | ord | ord | nonsplit |
| $\lambda$-invariant(s) | 6 | - | 2 | 1,1 | 3 | 5 | 3 | 1,1 | 1 | 1 | 1 | 1 | 1 | 3 | 1 | 1 |
| $\mu$-invariant(s) | 2 | - | 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.