Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-29304x+1920996\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-29304xz^2+1920996z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2373651x+1407527010\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(66, 528)$ | $1.0672116891040326697540416633$ | $\infty$ |
| $(102, 60)$ | $1.3046548642484262427370689673$ | $\infty$ |
| $(98, 0)$ | $0$ | $2$ |
| $(99, 0)$ | $0$ | $2$ |
Integral points
\( \left(-198, 0\right) \), \((-190,\pm 816)\), \((-144,\pm 1782)\), \((0,\pm 1386)\), \((24,\pm 1110)\), \((66,\pm 528)\), \((90,\pm 144)\), \((96,\pm 42)\), \( \left(98, 0\right) \), \( \left(99, 0\right) \), \((102,\pm 60)\), \((135,\pm 666)\), \((198,\pm 1980)\), \((246,\pm 3108)\), \((627,\pm 15180)\), \((774,\pm 21060)\), \((1430,\pm 53724)\), \((10791,\pm 1120878)\)
Invariants
| Conductor: | $N$ | = | \( 19536 \) | = | $2^{4} \cdot 3 \cdot 11 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $123656315904$ | = | $2^{10} \cdot 3^{6} \cdot 11^{2} \cdot 37^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2717811254229988}{120758121} \) | = | $2^{2} \cdot 3^{-6} \cdot 7^{3} \cdot 11^{-2} \cdot 19^{3} \cdot 37^{-2} \cdot 661^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2057096820537371630426376770$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.62808703158711607186161090912$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9418522684727934$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.298584417685089$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.2212849715670262831924813343$ |
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| Real period: | $\Omega$ | ≈ | $0.98316433737676607649158092506$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2^{2}\cdot( 2 \cdot 3 )\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.2043429789133879627517036854 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.204342979 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.983164 \cdot 1.221285 \cdot 96}{4^2} \\ & \approx 7.204342979\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 49152 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{2}^{*}$ | additive | 1 | 4 | 10 | 0 |
| $3$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $11$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $37$ | $2$ | $I_{2}$ | split 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 |
|---|---|---|
| $2$ | 2Cs | 4.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9768 = 2^{3} \cdot 3 \cdot 11 \cdot 37 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 2441 & 9766 \\ 0 & 9767 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6217 & 4 \\ 2666 & 9 \end{array}\right),\left(\begin{array}{rr} 3257 & 4 \\ 6514 & 9 \end{array}\right),\left(\begin{array}{rr} 9765 & 4 \\ 9764 & 5 \end{array}\right),\left(\begin{array}{rr} 3961 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4885 & 2 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9768])$ is a degree-$36944982835200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9768\Z)$.
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 | $2$ | \( 1 \) |
| $3$ | split multiplicative | $4$ | \( 6512 = 2^{4} \cdot 11 \cdot 37 \) |
| $11$ | split multiplicative | $12$ | \( 1776 = 2^{4} \cdot 3 \cdot 37 \) |
| $37$ | split multiplicative | $38$ | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 19536p
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 9768n2, its twist by $-4$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(i, \sqrt{74})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-33}, \sqrt{-74})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | ord | ord | split | ord | ord | ord | ss | ord | ss | split | ord | ord | ord |
| $\lambda$-invariant(s) | - | 5 | 2 | 2 | 3 | 2 | 2 | 2 | 2,2 | 2 | 2,2 | 3 | 2 | 2 | 2 |
| $\mu$-invariant(s) | - | 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.