Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-26248x-275692\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-26248xz^2-275692z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2126115x-194601150\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-22, 540)$ | $0.62131774897462287619709427733$ | $\infty$ |
| $(-76, 1134)$ | $0.63757600874096341616837889777$ | $\infty$ |
| $(-157, 0)$ | $0$ | $2$ |
Integral points
\( \left(-157, 0\right) \), \((-142,\pm 780)\), \((-118,\pm 1092)\), \((-92,\pm 1170)\), \((-76,\pm 1134)\), \((-22,\pm 540)\), \((-13,\pm 252)\), \((167,\pm 162)\), \((194,\pm 1404)\), \((218,\pm 2100)\), \((284,\pm 3906)\), \((428,\pm 8190)\), \((572,\pm 13122)\), \((1058,\pm 34020)\), \((1754,\pm 73164)\), \((3002,\pm 164268)\), \((6343,\pm 505050)\), \((7778,\pm 685860)\), \((20903,\pm 3022110)\)
Invariants
| Conductor: | $N$ | = | \( 109200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $1126620907776000$ | = | $2^{11} \cdot 3^{12} \cdot 5^{3} \cdot 7^{2} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{7812480469498}{4400862921} \) | = | $2 \cdot 3^{-12} \cdot 7^{-2} \cdot 13^{-2} \cdot 15749^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5782442997273075168363283507$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.54049990610549922288700907272$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9843288966942432$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.632437512903068$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.35216797941746610024527186975$ |
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| Real period: | $\Omega$ | ≈ | $0.40382252224525571999788842810$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 384 $ = $ 2^{2}\cdot( 2^{2} \cdot 3 )\cdot2\cdot2\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $13.652482723428140429067139776 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 13.652482723 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.403823 \cdot 0.352168 \cdot 384}{2^2} \\ & \approx 13.652482723\end{aligned}$$
Modular invariants
Modular form 109200.2.a.fs
For more coefficients, see the Downloads section to the right.
| Modular degree: | 589824 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{3}^{*}$ | additive | 1 | 4 | 11 | 0 |
| $3$ | $12$ | $I_{12}$ | split multiplicative | -1 | 1 | 12 | 12 |
| $5$ | $2$ | $III$ | additive | -1 | 2 | 3 | 0 |
| $7$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $13$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
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 \( 520 = 2^{3} \cdot 5 \cdot 13 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 41 & 4 \\ 82 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 457 & 66 \\ 64 & 455 \end{array}\right),\left(\begin{array}{rr} 2 & 1 \\ 259 & 0 \end{array}\right),\left(\begin{array}{rr} 517 & 4 \\ 516 & 5 \end{array}\right),\left(\begin{array}{rr} 108 & 1 \\ 103 & 0 \end{array}\right)$.
The torsion field $K:=\Q(E[520])$ is a degree-$1610219520$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/520\Z)$.
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 | $4$ | \( 5 \) |
| $3$ | split multiplicative | $4$ | \( 36400 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $10$ | \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | split multiplicative | $8$ | \( 15600 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 109200cl
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 54600n2, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $4$ | 4.0.169000.2 | \(\Z/4\Z\) | not in database |
| $8$ | 8.0.1827904000000.17 | \(\Z/2\Z \oplus \Z/4\Z\) | 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 | add | split | add | split | ord | nonsplit | ord | ord | ord | ord | ss | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 3 | - | 3 | 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 |
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.