Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-6673x-130455\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-6673xz^2-130455z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-8648235x-6060563802\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(278, 4279)$ | $0.99888777881789851849142104642$ | $\infty$ |
| $(90, -45)$ | $0$ | $2$ |
Integral points
\( \left(90, -45\right) \), \( \left(94, 227\right) \), \( \left(94, -321\right) \), \( \left(184, 2117\right) \), \( \left(184, -2301\right) \), \( \left(278, 4279\right) \), \( \left(278, -4557\right) \), \( \left(4508, 300379\right) \), \( \left(4508, -304887\right) \)
Invariants
| Conductor: | $N$ | = | \( 75106 \) | = | $2 \cdot 17 \cdot 47^{2}$ |
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| Discriminant: | $\Delta$ | = | $11727786277952$ | = | $2^{6} \cdot 17 \cdot 47^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{3048625}{1088} \) | = | $2^{-6} \cdot 5^{3} \cdot 17^{-1} \cdot 29^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.2086814643065673067584871570$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.71639233654846198665198817789$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9000957170016773$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.3875704180139063$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.99888777881789851849142104642$ |
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| Real period: | $\Omega$ | ≈ | $0.54383575781760796536865255904$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ ( 2 \cdot 3 )\cdot1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.2593853530090740625432506715 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.259385353 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.543836 \cdot 0.998888 \cdot 24}{2^2} \\ & \approx 3.259385353\end{aligned}$$
Modular invariants
Modular form 75106.2.a.a
For more coefficients, see the Downloads section to the right.
| Modular degree: | 211968 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| 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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
| $17$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $47$ | $4$ | $I_0^{*}$ | 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 | 8.6.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19176 = 2^{3} \cdot 3 \cdot 17 \cdot 47 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 9589 & 11844 \\ 15510 & 13537 \end{array}\right),\left(\begin{array}{rr} 16727 & 0 \\ 0 & 19175 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1505 & 17954 \\ 18894 & 11845 \end{array}\right),\left(\begin{array}{rr} 3291 & 3760 \\ 6110 & 11093 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 14242 & 7755 \\ 15933 & 15088 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19165 & 12 \\ 19164 & 13 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 19126 & 19167 \end{array}\right)$.
The torsion field $K:=\Q(E[19176])$ is a degree-$287195327889408$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19176\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 37553 = 17 \cdot 47^{2} \) |
| $3$ | good | $2$ | \( 37553 = 17 \cdot 47^{2} \) |
| $17$ | nonsplit multiplicative | $18$ | \( 4418 = 2 \cdot 47^{2} \) |
| $47$ | additive | $1106$ | \( 34 = 2 \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 75106.a
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 34.a4, its twist by $-47$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{17}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-47}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.4.2403392.2 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{17}, \sqrt{-47})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.234127821141.2 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.1669348707536896.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.5776293105664.5 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.0.12388383051417075058066236231810183168.1 | \(\Z/18\Z\) | 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 | split | ord | ss | ord | ord | ord | nonsplit | ord | ss | ss | ord | ord | ord | ord | add |
| $\lambda$-invariant(s) | 8 | 5 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1,1 | 1 | 1 | 1 | 1 | - |
| $\mu$-invariant(s) | 0 | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.