Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-14014242x-34489230084\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-14014242xz^2-34489230084z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-224227875x-2207534953250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(\frac{18531}{4}, -\frac{18531}{8}\right) \) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([37062:-18531:8]\) | $0$ | $2$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(18530, 0\right) \) | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 130050 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-337818919867201081125000$ | = | $-1 \cdot 2^{3} \cdot 3^{18} \cdot 5^{6} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{1107111813625}{1228691592} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-12} \cdot 5^{3} \cdot 17^{-2} \cdot 2069^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.2093255392065505366819408942$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.43869376662733746355917130019$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0188378451820035$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.2694370742466665$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.037389882029476824108987668125$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{2}\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.6920715061223313358471121050 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.692071506 \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{9 \cdot 0.037390 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 2.692071506\end{aligned}$$
Modular invariants
Modular form 130050.2.a.cu
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15925248 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $4$ | $I_{12}^{*}$ | additive | -1 | 2 | 18 | 12 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $17$ | $4$ | $I_{2}^{*}$ | additive | 1 | 2 | 8 | 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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2B | 8.6.0.5 | $6$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1759 & 1630 \\ 690 & 1619 \end{array}\right),\left(\begin{array}{rr} 1223 & 0 \\ 0 & 2039 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 1990 & 2031 \end{array}\right),\left(\begin{array}{rr} 826 & 1635 \\ 585 & 1216 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1799 & 1620 \\ 1410 & 1559 \end{array}\right),\left(\begin{array}{rr} 2029 & 12 \\ 2028 & 13 \end{array}\right),\left(\begin{array}{rr} 426 & 505 \\ 1445 & 1106 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$28877783040$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2040\Z)$.
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 |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 65025 = 3^{2} \cdot 5^{2} \cdot 17^{2} \) |
| $3$ | additive | $2$ | \( 7225 = 5^{2} \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $162$ | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 130050fv
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 102c2, its twist by $-255$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-255}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.2.2080800.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-2}, \sqrt{-255})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.1597339125.1 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.17734568509440000.211 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.277102632960000.207 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4329728640000.24 | \(\Z/12\Z\) | 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 |
| $18$ | 18.0.107539353849803643879988958671974609375.1 | \(\Z/18\Z\) | 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 | 17 |
|---|---|---|---|---|
| Reduction type | nonsplit | add | add | add |
| $\lambda$-invariant(s) | 8 | - | - | - |
| $\mu$-invariant(s) | 1 | - | - | - |
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 $0$.