Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-1688450x-845137500\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-1688450xz^2-845137500z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2188231875x-39397911725250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1500, -750)$ | $0$ | $2$ |
Integral points
\( \left(1500, -750\right) \)
Invariants
| Conductor: | $N$ | = | \( 61050 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 11 \cdot 37$ |
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| Discriminant: | $\Delta$ | = | $21167242874250000$ | = | $2^{4} \cdot 3 \cdot 5^{6} \cdot 11 \cdot 37^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{34069730739753390625}{1354703543952} \) | = | $2^{-4} \cdot 3^{-1} \cdot 5^{6} \cdot 11^{-1} \cdot 31^{3} \cdot 37^{-6} \cdot 47^{3} \cdot 89^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2158989992295865369413576309$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4111800430125363496409779643$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0298767255440278$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.957740723452023$ | |||
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: | $\Omega$ | ≈ | $0.13244284652071081758873715480$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2\cdot1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.3839712373727947165972687864 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $9$ = $3^2$ (exact) |
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BSD formula
$$\begin{aligned} 2.383971237 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{9 \cdot 0.132443 \cdot 1.000000 \cdot 8}{2^2} \\ & \approx 2.383971237\end{aligned}$$
Modular invariants
Modular form 61050.2.a.u
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1492992 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $3$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $37$ | $2$ | $I_{6}$ | nonsplit multiplicative | 1 | 1 | 6 | 6 |
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 | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 24420 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 37 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 11 & 2 \\ 24370 & 24411 \end{array}\right),\left(\begin{array}{rr} 24409 & 12 \\ 24408 & 13 \end{array}\right),\left(\begin{array}{rr} 3961 & 9780 \\ 4230 & 9841 \end{array}\right),\left(\begin{array}{rr} 17511 & 4480 \\ 12590 & 19121 \end{array}\right),\left(\begin{array}{rr} 4883 & 0 \\ 0 & 24419 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 8446 & 14655 \\ 7965 & 4876 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3266 & 14655 \\ 22765 & 4876 \end{array}\right)$.
The torsion field $K:=\Q(E[24420])$ is a degree-$554174742528000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/24420\Z)$.
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 | $4$ | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 550 = 2 \cdot 5^{2} \cdot 11 \) |
| $5$ | additive | $14$ | \( 2442 = 2 \cdot 3 \cdot 11 \cdot 37 \) |
| $11$ | split multiplicative | $12$ | \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \) |
| $37$ | nonsplit multiplicative | $38$ | \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 61050i
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2442i3, its twist by $5$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.4.1129425.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-15}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.2.1729072818000.27 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.1389129304550625.1 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.11480407475625.6 | \(\Z/12\Z\) | 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 |
| $18$ | 18.0.435803858150473769051826894545856572127133500000000.1 | \(\Z/18\Z\) | 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 | 11 | 37 |
|---|---|---|---|---|---|
| Reduction type | nonsplit | nonsplit | add | split | nonsplit |
| $\lambda$-invariant(s) | 5 | 2 | - | 1 | 0 |
| $\mu$-invariant(s) | 0 | 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 $0$.