Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-42950367x+121466910541\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-42950367xz^2+121466910541z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-687205875x+7773195068750\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 130050 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-1301843756669184000000000$ | = | $-1 \cdot 2^{17} \cdot 3^{6} \cdot 5^{9} \cdot 17^{8} $ |
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| j-invariant: | $j$ | = | \( -\frac{882216989}{131072} \) | = | $-1 \cdot 2^{-17} \cdot 17 \cdot 373^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3570158268458197572467761061$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.28817764785128775623443909086$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9569005082223869$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.4832345637879225$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
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: | $\Omega$ | ≈ | $0.083001656255407768394308505848$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $1.3280265000865242943089360936 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $4$ = $2^2$ (exact) |
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BSD formula
$$\begin{aligned} 1.328026500 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{4 \cdot 0.083002 \cdot 1.000000 \cdot 4}{1^2} \\ & \approx 1.328026500\end{aligned}$$
Modular invariants
Modular form 130050.2.a.o
For more coefficients, see the Downloads section to the right.
| Modular degree: | 24969600 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{17}$ | nonsplit multiplicative | 1 | 1 | 17 | 17 |
| $3$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $17$ | $1$ | $IV^{*}$ | additive | -1 | 2 | 8 | 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 | $\ell$-adic index |
|---|---|---|---|
| $17$ | 17B.4.2 | 17.72.1.2 | $72$ |
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 $576$, genus $17$, and generators
$\left(\begin{array}{rr} 1 & 1530 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 599 & 918 \\ 663 & 149 \end{array}\right),\left(\begin{array}{rr} 1 & 702 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 817 & 0 \\ 0 & 1633 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1360 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1530 & 1 \end{array}\right),\left(\begin{array}{rr} 1919 & 1224 \\ 1224 & 143 \end{array}\right),\left(\begin{array}{rr} 679 & 0 \\ 0 & 2039 \end{array}\right),\left(\begin{array}{rr} 103 & 714 \\ 1479 & 1939 \end{array}\right),\left(\begin{array}{rr} 1225 & 816 \\ 1224 & 1225 \end{array}\right),\left(\begin{array}{rr} 1 & 816 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 256 & 1785 \\ 1479 & 1 \end{array}\right),\left(\begin{array}{rr} 1429 & 306 \\ 765 & 1939 \end{array}\right),\left(\begin{array}{rr} 511 & 510 \\ 1530 & 1531 \end{array}\right),\left(\begin{array}{rr} 681 & 1360 \\ 680 & 681 \end{array}\right)$.
The torsion field $K:=\Q(E[2040])$ is a degree-$4812963840$ 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 | $4$ | \( 13005 = 3^{2} \cdot 5 \cdot 17^{2} \) |
| $3$ | additive | $6$ | \( 14450 = 2 \cdot 5^{2} \cdot 17^{2} \) |
| $5$ | additive | $14$ | \( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \) |
| $17$ | additive | $114$ | \( 225 = 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
17.
Its isogeny class 130050dq
consists of 2 curves linked by isogenies of
degree 17.
Twists
The minimal quadratic twist of this elliptic curve is 14450n1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.11560.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.5345344000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.45665106750000.8 | \(\Z/3\Z\) | not in database |
| $8$ | 8.8.519334883015625.1 | \(\Z/17\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | 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 | nonsplit | add | add | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | - | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.