Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-51192075x+140934735250\)
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(homogenize, simplify) |
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\(y^2z=x^3-51192075xz^2+140934735250z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-51192075x+140934735250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(59105/16, 3027375/64)$ | $5.1494736876215490450225432126$ | $\infty$ |
| $(4190, 0)$ | $0$ | $2$ |
Integral points
\( \left(4190, 0\right) \)
Invariants
| Conductor: | $N$ | = | \( 93600 \) | = | $2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 13$ |
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| Discriminant: | $\Delta$ | = | $5303399073921000000000$ | = | $2^{9} \cdot 3^{22} \cdot 5^{9} \cdot 13^{2} $ |
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| j-invariant: | $j$ | = | \( \frac{2543984126301795848}{909361981125} \) | = | $2^{3} \cdot 3^{-16} \cdot 5^{-3} \cdot 11^{6} \cdot 13^{-2} \cdot 5641^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.1378166231036875101344478814$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.2639311371326234950735215052$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0847562556631796$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.66682130897611$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $5.1494736876215490450225432126$ |
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| Real period: | $\Omega$ | ≈ | $0.13334566886736179262462030057$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 32 $ = $ 1\cdot2^{2}\cdot2^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4932801055262041389762711984 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.493280106 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.133346 \cdot 5.149474 \cdot 32}{2^2} \\ & \approx 5.493280106\end{aligned}$$
Modular invariants
Modular form 93600.2.a.k
For more coefficients, see the Downloads section to the right.
| Modular degree: | 9437184 |
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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 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_0^{*}$ | additive | -1 | 5 | 9 | 0 |
| $3$ | $4$ | $I_{16}^{*}$ | additive | -1 | 2 | 22 | 16 |
| $5$ | $4$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
| $13$ | $2$ | $I_{2}$ | 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 | 8.12.0.10 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \), index $192$, genus $3$, and generators
$\left(\begin{array}{rr} 1039 & 0 \\ 0 & 3119 \end{array}\right),\left(\begin{array}{rr} 1834 & 783 \\ 93 & 1060 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 15 & 166 \\ 2834 & 3075 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 8 & 65 \end{array}\right),\left(\begin{array}{rr} 2641 & 1056 \\ 1368 & 2209 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 2333 & 2850 \\ 768 & 2267 \end{array}\right),\left(\begin{array}{rr} 1442 & 3117 \\ 1779 & 2060 \end{array}\right),\left(\begin{array}{rr} 3105 & 16 \\ 3104 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[3120])$ is a degree-$77290536960$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3120\Z)$.
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 | $4$ | \( 225 = 3^{2} \cdot 5^{2} \) |
| $3$ | additive | $8$ | \( 10400 = 2^{5} \cdot 5^{2} \cdot 13 \) |
| $5$ | additive | $18$ | \( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \) |
| $13$ | nonsplit multiplicative | $14$ | \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 93600.k
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 6240.p1, its twist by $-15$.
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 |
| $2$ | \(\Q(\sqrt{6}) \) | \(\Z/4\Z\) | not in database |
| $2$ | \(\Q(\sqrt{15}) \) | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{6}, \sqrt{10})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | 8.8.21233664000000.3 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.379034173440000.213 | \(\Z/8\Z\) | 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 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | add | ord | ss | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 3 | 3,3 | 3 | 3 | 1 | 1 | 1 | 1 | 1 | 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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.