Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-56050954x-103141949044\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-56050954xz^2-103141949044z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-72642035763x-4811972848477938\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-1747148707775/279458089, 267409060681298921/4671700873813)$ | $24.785495950432452460509609128$ | $\infty$ |
| $(-7913/4, 7909/8)$ | $0$ | $2$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 68970 \) | = | $2 \cdot 3 \cdot 5 \cdot 11^{2} \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $6674818853210449218750000$ | = | $2^{4} \cdot 3^{2} \cdot 5^{18} \cdot 11^{6} \cdot 19^{3} $ |
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| j-invariant: | $j$ | = | \( \frac{10993009831928446009969}{3767761230468750000} \) | = | $2^{-4} \cdot 3^{-2} \cdot 5^{-18} \cdot 19^{-3} \cdot 23^{3} \cdot 47^{3} \cdot 67^{3} \cdot 307^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.4653070574848057578703858541$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.2663594210856204858394140651$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0531544430296054$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.84655078184611$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $24.785495950432452460509609128$ |
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| Real period: | $\Omega$ | ≈ | $0.056714425873300650012814762321$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 16 $ = $ 2\cdot2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.6227806912551790681611457725 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.622780691 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.056714 \cdot 24.785496 \cdot 16}{2^2} \\ & \approx 5.622780691\end{aligned}$$
Modular invariants
Modular form 68970.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 18662400 |
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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 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{18}$ | nonsplit multiplicative | 1 | 1 | 18 | 18 |
| $11$ | $2$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 0 |
| $19$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
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 \( 12540 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5226 & 4477 \\ 5225 & 1046 \end{array}\right),\left(\begin{array}{rr} 3419 & 0 \\ 0 & 12539 \end{array}\right),\left(\begin{array}{rr} 4181 & 3432 \\ 12166 & 8053 \end{array}\right),\left(\begin{array}{rr} 12529 & 12 \\ 12528 & 13 \end{array}\right),\left(\begin{array}{rr} 5017 & 3432 \\ 462 & 8053 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 12490 & 12531 \end{array}\right),\left(\begin{array}{rr} 4390 & 10263 \\ 3993 & 6832 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[12540])$ is a degree-$37444239360000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/12540\Z)$.
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 | $4$ | \( 2299 = 11^{2} \cdot 19 \) |
| $3$ | split multiplicative | $4$ | \( 242 = 2 \cdot 11^{2} \) |
| $5$ | nonsplit multiplicative | $6$ | \( 13794 = 2 \cdot 3 \cdot 11^{2} \cdot 19 \) |
| $11$ | additive | $62$ | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| $19$ | nonsplit multiplicative | $20$ | \( 3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 68970x
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 570k4, its twist by $-11$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{19}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{33}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.2069100.3 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{19}, \sqrt{33})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.1257507504.3 | \(\Z/6\Z\) | not in database |
| $8$ | 8.0.24728065702560000.81 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.0.4281174810000.9 | \(\Z/12\Z\) | 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 |
| $18$ | 18.6.825232340558978998557500385385931652786432.3 | \(\Z/18\Z\) | 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 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | nonsplit | split | nonsplit | ord | add | ord | ord | nonsplit | ss | ss | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 3 | 2 | 3 | 1 | - | 1 | 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 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.