Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3-x^2-736x-18020\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3-x^2z-736xz^2-18020z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-954288x-852179184\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(110, 1104\right) \) | $1.0073138355381507854903037036$ | $\infty$ |
| \( \left(\frac{205}{4}, \frac{2205}{8}\right) \) | $1.7886686689055144021605595583$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([110:1104:1]\) | $1.0073138355381507854903037036$ | $\infty$ |
| \([410:2205:8]\) | $1.7886686689055144021605595583$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(3948, 238572\right) \) | $1.0073138355381507854903037036$ | $\infty$ |
| \( \left(1833, 59643\right) \) | $1.7886686689055144021605595583$ | $\infty$ |
Integral points
\( \left(37, 63\right) \), \( \left(37, -64\right) \), \( \left(110, 1104\right) \), \( \left(110, -1105\right) \), \( \left(392, 7731\right) \), \( \left(392, -7732\right) \)
\([37:63:1]\), \([37:-64:1]\), \([110:1104:1]\), \([110:-1105:1]\), \([392:7731:1]\), \([392:-7732:1]\)
\((1320,\pm 13716)\), \((3948,\pm 238572)\), \((14100,\pm 1670004)\)
Invariants
| Conductor: | $N$ | = | \( 24299 \) | = | $11 \cdot 47^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-118571368619$ | = | $-1 \cdot 11 \cdot 47^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{4096}{11} \) | = | $-1 \cdot 2^{12} \cdot 11^{-1}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.81234500351957604142108140681$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.1127287973354532519893939281$ |
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| $abc$ quality: | $Q$ | ≈ | $0.8254556483942886$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2826323419051033$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $1.7028589088340650874922104335$ |
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| Real period: | $\Omega$ | ≈ | $0.42558054685344181147329257443$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 1\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $2.8988145025434265406930062827 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.898814503 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.425581 \cdot 1.702859 \cdot 4}{1^2} \\ & \approx 2.898814503\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 20608 |
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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 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $47$ | $4$ | $I_0^{*}$ | 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 | $\ell$-adic index |
|---|---|---|---|
| $5$ | 5B.4.1 | 25.60.0.1 | $60$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 25850 = 2 \cdot 5^{2} \cdot 11 \cdot 47 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 38 & 41 \\ 23291 & 23089 \end{array}\right),\left(\begin{array}{rr} 25616 & 1645 \\ 20727 & 4137 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 11516 & 24205 \\ 14335 & 16686 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 25801 & 50 \\ 25800 & 51 \end{array}\right),\left(\begin{array}{rr} 3849 & 0 \\ 0 & 25849 \end{array}\right)$.
The torsion field $K:=\Q(E[25850])$ is a degree-$94519180800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/25850\Z)$.
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 |
|---|---|---|---|
| $11$ | nonsplit multiplicative | $12$ | \( 2209 = 47^{2} \) |
| $47$ | additive | $1106$ | \( 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 24299c
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 11a3, its twist by $-47$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-47}) \) | \(\Z/5\Z\) | 2.0.47.1-121.1-a3 |
| $3$ | 3.1.44.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.21296.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.201001328.2 | \(\Z/10\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.49162139063457167.3 | \(\Z/25\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | not in database |
| $16$ | deg 16 | \(\Z/15\Z\) | not in database |
| $20$ | 20.4.73758420327597080481788270375545928955078125.1 | \(\Z/5\Z\) | 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 | nonsplit | ord | ord | ss | ord | ss | ord | ord | ord | ord | add |
| $\lambda$-invariant(s) | 8,9 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 | 2 | 2,2 | 2 | 2 | 2 | 2 | - |
| $\mu$-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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.