Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-500x+50650\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-500xz^2+50650z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-648675x+2372853150\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(59, 451)$ | $1.0237247010602927295393298139$ | $\infty$ |
Integral points
\( \left(-9, 238\right) \), \( \left(-9, -229\right) \), \( \left(59, 451\right) \), \( \left(59, -510\right) \)
Invariants
| Conductor: | $N$ | = | \( 48050 \) | = | $2 \cdot 5^{2} \cdot 31^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1109379601250$ | = | $-1 \cdot 2 \cdot 5^{4} \cdot 31^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{25}{2} \) | = | $-1 \cdot 2^{-1} \cdot 5^{2}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.99089400609869159021075678227$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2625789002885816576207451577$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0904350906962674$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2650053994862502$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.0237247010602927295393298139$ |
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| Real period: | $\Omega$ | ≈ | $0.71733162833271108979684427428$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 6 $ = $ 1\cdot3\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $4.4061006406559860246723785638 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 4.406100641 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.717332 \cdot 1.023725 \cdot 6}{1^2} \\ & \approx 4.406100641\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 61200 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $5$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $31$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $2$ | 2G | 8.2.0.1 |
| $3$ | 3B | 3.4.0.1 |
| $5$ | 5B.4.2 | 5.12.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \), index $384$, genus $9$, and generators
$\left(\begin{array}{rr} 1 & 2232 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 3162 \\ 930 & 1861 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2790 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1860 & 1 \end{array}\right),\left(\begin{array}{rr} 961 & 2760 \\ 960 & 961 \end{array}\right),\left(\begin{array}{rr} 1 & 1116 \\ 1860 & 1 \end{array}\right),\left(\begin{array}{rr} 1919 & 0 \\ 0 & 3719 \end{array}\right),\left(\begin{array}{rr} 1241 & 2480 \\ 1240 & 2481 \end{array}\right),\left(\begin{array}{rr} 2481 & 2170 \\ 1240 & 2481 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2760 & 1 \end{array}\right),\left(\begin{array}{rr} 1861 & 930 \\ 2325 & 931 \end{array}\right),\left(\begin{array}{rr} 2791 & 930 \\ 3255 & 931 \end{array}\right),\left(\begin{array}{rr} 2233 & 930 \\ 0 & 1489 \end{array}\right),\left(\begin{array}{rr} 2326 & 465 \\ 2325 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3720])$ is a degree-$82280448000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3720\Z)$.
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 | $4$ | \( 24025 = 5^{2} \cdot 31^{2} \) |
| $5$ | additive | $14$ | \( 1922 = 2 \cdot 31^{2} \) |
| $31$ | additive | $482$ | \( 50 = 2 \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 5 and 15.
Its isogeny class 48050l
consists of 4 curves linked by isogenies of
degrees dividing 15.
Twists
The minimal quadratic twist of this elliptic curve is 50a1, its twist by $-31$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-31}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.200.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.4.120125.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.320000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.8043570000.4 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.1191640000.4 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.14430015625.1 | \(\Z/15\Z\) | not in database |
| $10$ | 10.0.357864387500000000.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.640203209153899711757437500000000.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.8526415183683375808512000000000000.1 | \(\Z/6\Z\) | not in database |
| $20$ | 20.0.128066919840750156250000000000000000.1 | \(\Z/15\Z\) | 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 | ord | add | ord | ord | ord | ord | ord | ord | ss | add | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 18 | 3 | - | 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 |
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.