Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3+x^2-28325x-1830375\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3+x^2z-28325xz^2-1830375z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-36709875x-84847331250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(216, 1371)$ | $3.9842893395221901806141148471$ | $\infty$ |
| $(-90, 45)$ | $0$ | $2$ |
Integral points
\( \left(-90, 45\right) \), \( \left(216, 1371\right) \), \( \left(216, -1587\right) \)
Invariants
| Conductor: | $N$ | = | \( 119850 \) | = | $2 \cdot 3 \cdot 5^{2} \cdot 17 \cdot 47$ |
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| Discriminant: | $\Delta$ | = | $25786476562500$ | = | $2^{2} \cdot 3^{5} \cdot 5^{9} \cdot 17^{2} \cdot 47 $ |
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| j-invariant: | $j$ | = | \( \frac{1286848396133}{13202676} \) | = | $2^{-2} \cdot 3^{-5} \cdot 17^{-2} \cdot 47^{-1} \cdot 73^{3} \cdot 149^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.3908357891936885022484862725$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.18375735486811322129791677258$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8577627765099891$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.6230692728259775$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.9842893395221901806141148471$ |
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| Real period: | $\Omega$ | ≈ | $0.36823424612622717159102787742$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2\cdot1\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.9343035625754345511104260545 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.934303563 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.368234 \cdot 3.984289 \cdot 8}{2^2} \\ & \approx 2.934303563\end{aligned}$$
Modular invariants
Modular form 119850.2.a.c
For more coefficients, see the Downloads section to the right.
| Modular degree: | 396800 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $3$ | $1$ | $I_{5}$ | nonsplit multiplicative | 1 | 1 | 5 | 5 |
| $5$ | $2$ | $III^{*}$ | additive | -1 | 2 | 9 | 0 |
| $17$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $47$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
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 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 95880 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \cdot 47 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 63922 & 1 \\ 63919 & 0 \end{array}\right),\left(\begin{array}{rr} 45121 & 4 \\ 90242 & 9 \end{array}\right),\left(\begin{array}{rr} 95877 & 4 \\ 95876 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 59929 & 35956 \\ 11984 & 83895 \end{array}\right),\left(\begin{array}{rr} 76708 & 1 \\ 76703 & 0 \end{array}\right),\left(\begin{array}{rr} 47941 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 81602 & 1 \\ 55079 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right)$.
The torsion field $K:=\Q(E[95880])$ is a degree-$1102830059095326720$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/95880\Z)$.
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 | $4$ | \( 705 = 3 \cdot 5 \cdot 47 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 39950 = 2 \cdot 5^{2} \cdot 17 \cdot 47 \) |
| $5$ | additive | $14$ | \( 1598 = 2 \cdot 17 \cdot 47 \) |
| $17$ | split multiplicative | $18$ | \( 7050 = 2 \cdot 3 \cdot 5^{2} \cdot 47 \) |
| $47$ | nonsplit multiplicative | $48$ | \( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 119850r
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 119850cs1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{705}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.325992000.1 | \(\Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | nonsplit | add | ord | ss | ord | split | ord | ord | ord | ord | ord | ord | ss | nonsplit |
| $\lambda$-invariant(s) | 5 | 3 | - | 1 | 1,1 | 1 | 2 | 1 | 1 | 1 | 5 | 1 | 1 | 1,1 | 1 |
| $\mu$-invariant(s) | 0 | 0 | - | 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.