Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-x^2-8704008x-10011913488\)
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(homogenize, simplify) |
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\(y^2z=x^3-x^2z-8704008xz^2-10011913488z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-705024675x-7300800006750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(3412, 0)$ | $0$ | $2$ |
Integral points
\( \left(3412, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 109200 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $-1125625028935680000000$ | = | $-1 \cdot 2^{44} \cdot 3^{2} \cdot 5^{7} \cdot 7 \cdot 13 $ |
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| j-invariant: | $j$ | = | \( -\frac{1139466686381936641}{17587891077120} \) | = | $-1 \cdot 2^{-32} \cdot 3^{-2} \cdot 5^{-1} \cdot 7^{-1} \cdot 13^{-1} \cdot 47^{3} \cdot 71^{3} \cdot 313^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.8415700678875410915287953030$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.3437039311105455948111835149$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9772300527284133$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.13560340405276$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.043907772191229777298280867451$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 2^{2}\cdot2\cdot2^{2}\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.35126217752983821838624693960 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.351262178 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.043908 \cdot 1.000000 \cdot 32}{2^2} \\ & \approx 0.351262178\end{aligned}$$
Modular invariants
Modular form 109200.2.a.e
For more coefficients, see the Downloads section to the right.
| Modular degree: | 4718592 |
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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 5 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $4$ | $I_{36}^{*}$ | additive | -1 | 4 | 44 | 32 |
| $3$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $5$ | $4$ | $I_{1}^{*}$ | additive | 1 | 2 | 7 | 1 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $13$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $2$ | 2B | 32.48.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 14560 = 2^{5} \cdot 5 \cdot 7 \cdot 13 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 5463 & 2 \\ 5434 & 14543 \end{array}\right),\left(\begin{array}{rr} 8159 & 14528 \\ 4086 & 14081 \end{array}\right),\left(\begin{array}{rr} 11626 & 14557 \\ 1555 & 212 \end{array}\right),\left(\begin{array}{rr} 8976 & 25 \\ 919 & 3186 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 11998 & 12555 \end{array}\right),\left(\begin{array}{rr} 2112 & 29 \\ 13227 & 2562 \end{array}\right),\left(\begin{array}{rr} 14529 & 32 \\ 14528 & 33 \end{array}\right)$.
The torsion field $K:=\Q(E[14560])$ is a degree-$12984810209280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/14560\Z)$.
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 | $2$ | \( 2275 = 5^{2} \cdot 7 \cdot 13 \) |
| $3$ | 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 | $8$ | \( 15600 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 13 \) |
| $13$ | 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, 4, 8 and 16.
Its isogeny class 109200cz
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 2730v1, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-455}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{91}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{91})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-7})\) | \(\Z/8\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-13})\) | \(\Z/8\Z\) | 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 |
| $8$ | 8.0.10971993760000.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.46356673636000000.42 | \(\Z/16\Z\) | not in database |
| $8$ | 8.0.19307236000000.2 | \(\Z/16\Z\) | 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 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/32\Z\) | 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 | 13 |
|---|---|---|---|---|---|
| Reduction type | add | nonsplit | add | nonsplit | nonsplit |
| $\lambda$-invariant(s) | - | 0 | - | 0 | 0 |
| $\mu$-invariant(s) | - | 0 | - | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.