Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-624408x+190119312\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-624408xz^2+190119312z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-50577075x+138445247250\)
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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 |
|---|---|---|
| $(432, 900)$ | $1.2707700309028299798995437014$ | $\infty$ |
| $(456, 12)$ | $1.8150992704327859921056329488$ | $\infty$ |
| $(457, 0)$ | $0$ | $2$ |
Integral points
\((-912,\pm 444)\), \((-768,\pm 14700)\), \((408,\pm 1764)\), \((432,\pm 900)\), \((456,\pm 12)\), \( \left(457, 0\right) \), \((482,\pm 950)\), \((506,\pm 1862)\), \((557,\pm 3850)\), \((3058,\pm 163914)\), \((11482,\pm 1227450)\)
Invariants
| Conductor: | $N$ | = | \( 109200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $44946720000000$ | = | $2^{11} \cdot 3^{2} \cdot 5^{7} \cdot 7^{4} \cdot 13 $ |
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| j-invariant: | $j$ | = | \( \frac{841356017734178}{1404585} \) | = | $2 \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-4} \cdot 13^{-1} \cdot 74929^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.8818934524830794359786847469$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.44178958075274604837917563562$ |
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| $abc$ quality: | $Q$ | ≈ | $0.936065372896473$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.451992576278156$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.1771592780808618567701879906$ |
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| Real period: | $\Omega$ | ≈ | $0.54630563716000664879328659315$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2\cdot2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $9.5151550928862826731741612636 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 9.515155093 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.546306 \cdot 2.177159 \cdot 32}{2^2} \\ & \approx 9.515155093\end{aligned}$$
Modular invariants
Modular form 109200.2.a.g
For more coefficients, see the Downloads section to the right.
| Modular degree: | 983040 |
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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 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $2$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $2$ | $I_{4}$ | nonsplit multiplicative | 1 | 1 | 4 | 4 |
| $13$ | $1$ | $I_{1}$ | split 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 |
|---|---|---|
| $2$ | 2B | 4.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3640 = 2^{3} \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 3181 & 3184 \\ 1342 & 3179 \end{array}\right),\left(\begin{array}{rr} 456 & 2283 \\ 1369 & 1398 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 3634 & 3635 \end{array}\right),\left(\begin{array}{rr} 2244 & 1 \\ 1983 & 6 \end{array}\right),\left(\begin{array}{rr} 521 & 8 \\ 2084 & 33 \end{array}\right),\left(\begin{array}{rr} 3633 & 8 \\ 3632 & 9 \end{array}\right),\left(\begin{array}{rr} 1452 & 3639 \\ 2889 & 3634 \end{array}\right)$.
The torsion field $K:=\Q(E[3640])$ is a degree-$811550638080$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3640\Z)$.
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 | $4$ | \( 325 = 5^{2} \cdot 13 \) |
| $3$ | nonsplit multiplicative | $4$ | \( 36400 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 13 \) |
| $5$ | additive | $18$ | \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 15600 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | split 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 and 4.
Its isogeny class 109200.g
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 10920.c1, its twist by $-20$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{130}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{10}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{13}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{10}, \sqrt{13})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/8\Z\) | not in database |
| $8$ | deg 8 | \(\Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/12\Z\) | 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 | nonsplit | add | nonsplit | ord | split | ord | ord | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 6 | - | 2 | 2 | 3 | 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 |
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.