Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-448x+3508\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-448xz^2+3508z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-36315x+2666250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(18, 40)$ | $0.22035584825563752770771466299$ | $\infty$ |
Integral points
\((-21,\pm 64)\), \((12,\pm 2)\), \((18,\pm 40)\)
Invariants
| Conductor: | $N$ | = | \( 4400 \) | = | $2^{4} \cdot 5^{2} \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-11264000$ | = | $-1 \cdot 2^{13} \cdot 5^{3} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( -\frac{19465109}{22} \) | = | $-1 \cdot 2^{-1} \cdot 11^{-1} \cdot 269^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.26683732912770937591308982963$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.82866932954076102715433212513$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0648872065191846$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.567872378069047$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.22035584825563752770771466299$ |
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| Real period: | $\Omega$ | ≈ | $2.2610916432387235563842557777$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 8 $ = $ 2^{2}\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $3.9859581362368181864219981329 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 3.985958136 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 2.261092 \cdot 0.220356 \cdot 8}{1^2} \\ & \approx 3.985958136\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1152 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $11$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $5$ | 5B.4.2 | 25.60.0.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \), index $1200$, genus $37$, and generators
$\left(\begin{array}{rr} 1670 & 1209 \\ 787 & 2098 \end{array}\right),\left(\begin{array}{rr} 1101 & 50 \\ 1125 & 1251 \end{array}\right),\left(\begin{array}{rr} 2031 & 10 \\ 45 & 2021 \end{array}\right),\left(\begin{array}{rr} 1 & 50 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 38 & 41 \\ 1841 & 1639 \end{array}\right),\left(\begin{array}{rr} 2151 & 50 \\ 2150 & 51 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 50 & 1 \end{array}\right),\left(\begin{array}{rr} 1649 & 2150 \\ 0 & 2199 \end{array}\right)$.
The torsion field $K:=\Q(E[2200])$ is a degree-$5068800000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2200\Z)$.
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 |
|---|---|---|---|
| $2$ | additive | $4$ | \( 55 = 5 \cdot 11 \) |
| $11$ | nonsplit multiplicative | $12$ | \( 400 = 2^{4} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5 and 25.
Its isogeny class 4400v
consists of 3 curves linked by isogenies of
degrees dividing 25.
Twists
The minimal quadratic twist of this elliptic curve is 550k1, its twist by $-4$.
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 |
|---|---|---|---|
| $3$ | 3.1.440.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\zeta_{20})^+\) | \(\Z/5\Z\) | not in database |
| $6$ | 6.0.85184000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $8$ | deg 8 | \(\Z/3\Z\) | not in database |
| $10$ | 10.0.54875873536000000.2 | \(\Z/5\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/10\Z\) | not in database |
| $20$ | 20.20.1470391355634309152000000000000000.1 | \(\Z/25\Z\) | 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 | add | ord | add | ord | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 1 | - | 3 | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 3 | 1 | 1 | 1 |
| $\mu$-invariant(s) | - | 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.