Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+x^2-2823140x+1824832093\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+x^2z-2823140xz^2+1824832093z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-3658789872x+85183271619408\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(95429/100, 830883/1000)$ | $5.6128375836361512609643895874$ | $\infty$ |
Integral points
None
Invariants
| Conductor: | $N$ | = | \( 3971 \) | = | $11 \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-517504691$ | = | $-1 \cdot 11 \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{52893159101157376}{11} \) | = | $-1 \cdot 2^{12} \cdot 11^{-1} \cdot 29^{3} \cdot 809^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.9689286046818673526158791211$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.49670911509864712261136540516$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0929566831983986$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.77871624045694$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.6128375836361512609643895874$ |
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| Real period: | $\Omega$ | ≈ | $0.66935096951503013826141370190$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 1\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.5139165566745138133921444429 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.513916557 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.669351 \cdot 5.612838 \cdot 2}{1^2} \\ & \approx 7.513916557\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 36000 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $11$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
| $19$ | $2$ | $I_0^{*}$ | 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 |
|---|---|---|
| $5$ | 5B.4.2 | 25.60.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10450 = 2 \cdot 5^{2} \cdot 11 \cdot 19 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 6936 & 6745 \\ 1615 & 3516 \end{array}\right),\left(\begin{array}{rr} 1540 & 4959 \\ 7277 & 1863 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10401 & 50 \\ 10400 & 51 \end{array}\right),\left(\begin{array}{rr} 38 & 41 \\ 7891 & 7689 \end{array}\right),\left(\begin{array}{rr} 549 & 0 \\ 0 & 10449 \end{array}\right)$.
The torsion field $K:=\Q(E[10450])$ is a degree-$2437776000000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10450\Z)$.
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 |
|---|---|---|---|
| $11$ | split multiplicative | $12$ | \( 361 = 19^{2} \) |
| $19$ | additive | $182$ | \( 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 3971b
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 11a2, its twist by $-19$.
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 |
|---|---|---|---|
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | not in database |
| $4$ | 4.4.45125.1 | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | 8.2.4172861087307.1 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.5183337996925966796875.1 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $20$ | 20.20.819915551464126397076740040097502060234546661376953125.1 | \(\Z/25\Z\) | 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 | ss | ord | ord | ord | split | ord | ord | add | ord | ss | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2,3 | 1 | 3 | 1 | 2 | 1 | 1 | - | 3 | 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 |
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.