Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+x^2-226908338x-1313092090969\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+x^2z-226908338xz^2-1313092090969z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-294073206075x-61259213498150250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-36045/4, 36041/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 127050 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{2}$ |
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| Discriminant: | $\Delta$ | = | $2954990204694596289375000$ | = | $2^{3} \cdot 3^{12} \cdot 5^{7} \cdot 7^{3} \cdot 11^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{46676570542430835889}{106752955783320} \) | = | $2^{-3} \cdot 3^{-12} \cdot 5^{-1} \cdot 7^{-3} \cdot 11^{-4} \cdot 47^{3} \cdot 76607^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.5767820794836103785586824941$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.5731154868673749192273310385$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9957188233471589$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.89957376890871$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.038904212017220109633832954297$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 144 $ = $ 3\cdot2\cdot2\cdot3\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $5.6022065304796957872719454188 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 5.602206530 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.038904 \cdot 1.000000 \cdot 144}{2^2} \\ & \approx 5.602206530\end{aligned}$$
Modular invariants
Modular form 127050.2.a.gv
For more coefficients, see the Downloads section to the right.
| Modular degree: | 39813120 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $2$ | $I_{12}$ | nonsplit multiplicative | 1 | 1 | 12 | 12 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $4$ | $I_{4}^{*}$ | additive | -1 | 2 | 10 | 4 |
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 |
| $3$ | 3B | 3.4.0.1 |
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 $384$, genus $5$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 7934 & 11 \end{array}\right),\left(\begin{array}{rr} 6934 & 3 \\ 6975 & 34 \end{array}\right),\left(\begin{array}{rr} 1926 & 409 \\ 4235 & 2696 \end{array}\right),\left(\begin{array}{rr} 5528 & 9237 \\ 6003 & 86 \end{array}\right),\left(\begin{array}{rr} 9217 & 24 \\ 9216 & 25 \end{array}\right),\left(\begin{array}{rr} 4621 & 24 \\ 3092 & 1829 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6719 & 9216 \\ 6708 & 8951 \end{array}\right),\left(\begin{array}{rr} 1336 & 21 \\ 1035 & 8866 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$2452488192000$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 21175 = 5^{2} \cdot 7 \cdot 11^{2} \) |
| $3$ | nonsplit multiplicative | $4$ | \( 3025 = 5^{2} \cdot 11^{2} \) |
| $5$ | additive | $18$ | \( 5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2} \) |
| $7$ | split multiplicative | $8$ | \( 18150 = 2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \) |
| $11$ | additive | $72$ | \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 127050.gv
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2310.u4, its twist by $-55$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{70}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-385}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-22}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-22}, \sqrt{70})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-55}, \sqrt{70})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{7}, \sqrt{-55})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{-22})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.13588678125.1 | \(\Z/6\Z\) | 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 |
| $8$ | 8.0.1439868559360000.119 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.491913830421883710981936145089524110095000000000000.2 | \(\Z/18\Z\) | 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 |
|---|---|---|---|---|---|
| Reduction type | split | nonsplit | add | split | add |
| $\lambda$-invariant(s) | 5 | 2 | - | 3 | - |
| $\mu$-invariant(s) | 1 | 0 | - | 0 | - |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.