Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-5113008x-8450772012\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-5113008xz^2-8450772012z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-414153675x-6159370335750\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(2924, 40086)$ | $7.7086348043759812983449566790$ | $\infty$ |
| $(2843, 0)$ | $0$ | $2$ |
Integral points
\( \left(2843, 0\right) \), \((2924,\pm 40086)\)
Invariants
| Conductor: | $N$ | = | \( 92400 \) | = | $2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11$ |
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| Discriminant: | $\Delta$ | = | $-22284281250000000000000$ | = | $-1 \cdot 2^{13} \cdot 3^{3} \cdot 5^{18} \cdot 7^{4} \cdot 11 $ |
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| j-invariant: | $j$ | = | \( -\frac{230979395175477481}{348191894531250} \) | = | $-1 \cdot 2^{-1} \cdot 3^{-3} \cdot 5^{-12} \cdot 7^{-4} \cdot 11^{-1} \cdot 13^{3} \cdot 109^{3} \cdot 433^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.9798408907354568569847321110$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4819747539584613602671203229$ |
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| $abc$ quality: | $Q$ | ≈ | $0.9894412364238445$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.180899022649596$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $7.7086348043759812983449566790$ |
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| Real period: | $\Omega$ | ≈ | $0.047646755809355526282578142592$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 96 $ = $ 2\cdot3\cdot2^{2}\cdot2^{2}\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $8.8149945635424357077458849631 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 8.814994564 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.047647 \cdot 7.708635 \cdot 96}{2^2} \\ & \approx 8.814994564\end{aligned}$$
Modular invariants
Modular form 92400.2.a.hp
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5308416 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{5}^{*}$ | additive | -1 | 4 | 13 | 1 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $5$ | $4$ | $I_{12}^{*}$ | additive | 1 | 2 | 18 | 12 |
| $7$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
| $11$ | $1$ | $I_{1}$ | 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 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9240 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 4242 & 2719 \\ 7037 & 6692 \end{array}\right),\left(\begin{array}{rr} 3376 & 3 \\ 7101 & 9154 \end{array}\right),\left(\begin{array}{rr} 5543 & 9216 \\ 1836 & 8951 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 7934 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 16 & 4641 \\ 7415 & 5786 \end{array}\right),\left(\begin{array}{rr} 6914 & 9219 \\ 7215 & 374 \end{array}\right),\left(\begin{array}{rr} 9217 & 24 \\ 9216 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right),\left(\begin{array}{rr} 5281 & 24 \\ 7932 & 289 \end{array}\right)$.
The torsion field $K:=\Q(E[9240])$ is a degree-$2452488192000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9240\Z)$.
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 | $4$ | \( 825 = 3 \cdot 5^{2} \cdot 11 \) |
| $3$ | split multiplicative | $4$ | \( 30800 = 2^{4} \cdot 5^{2} \cdot 7 \cdot 11 \) |
| $5$ | additive | $18$ | \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \) |
| $7$ | split multiplicative | $8$ | \( 13200 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 11 \) |
| $11$ | split multiplicative | $12$ | \( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 92400hi
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 2310l5, 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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-66}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{30}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-55}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{30}, \sqrt{-55})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-66})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{-6})\) | \(\Z/12\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-5}, \sqrt{11})\) | \(\Z/12\Z\) | not in database |
| $6$ | 6.2.30372227424000.3 | \(\Z/6\Z\) | 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 |
| $8$ | 8.0.48575324160000.103 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | 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 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.32566674436760995497419322748032000000000000000.1 | \(\Z/18\Z\) | 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 | split | ord | ord | ord | ss | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | - | 4 | - | 2 | 2 | 1 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
| $\mu$-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.