Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3-2025x-30375\)
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(homogenize, simplify) |
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\(y^2z=x^3-2025xz^2-30375z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-2025x-30375\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 8100 \) | = | $2^{2} \cdot 3^{4} \cdot 5^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $132860250000$ | = | $2^{4} \cdot 3^{12} \cdot 5^{6} $ |
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| j-invariant: | $j$ | = | \( 6912 \) | = | $2^{8} \cdot 3^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $0.85967219885046060239479751576$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.2747081062213477127732380949$ |
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| $abc$ quality: | $Q$ | ≈ | $0.7737056144690831$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.828337251519447$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.71856403857274468206325104732$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 3\cdot1\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $2.1556921157182340461897531420 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 2.155692116 \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 0.718564 \cdot 1.000000 \cdot 3}{1^2} \\ & \approx 2.155692116\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 5832 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV$ | additive | -1 | 2 | 4 | 0 |
| $3$ | $1$ | $II^{*}$ | additive | 1 | 4 | 12 | 0 |
| $5$ | $1$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
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 | $\ell$-adic index |
|---|---|---|---|
| $2$ | 2Cn | 2.2.0.1 | $2$ |
| $3$ | 3B | 3.4.0.1 | $4$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 2 \\ 142 & 165 \end{array}\right),\left(\begin{array}{rr} 7 & 12 \\ 114 & 67 \end{array}\right),\left(\begin{array}{rr} 91 & 120 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 169 & 12 \\ 168 & 13 \end{array}\right),\left(\begin{array}{rr} 64 & 175 \\ 45 & 119 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right),\left(\begin{array}{rr} 143 & 0 \\ 0 & 179 \end{array}\right)$.
The torsion field $K:=\Q(E[180])$ is a degree-$1866240$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/180\Z)$.
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 | $2$ | \( 2025 = 3^{4} \cdot 5^{2} \) |
| $3$ | additive | $2$ | \( 100 = 2^{2} \cdot 5^{2} \) |
| $5$ | additive | $14$ | \( 324 = 2^{2} \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 8100b
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 324c1, 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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-15}) \) | \(\Z/3\Z\) | not in database |
| $3$ | \(\Q(\zeta_{9})^+\) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.1458000.2 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.2460375.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $12$ | 12.0.19131876000000.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $18$ | 18.0.2626055739156296621016000000000.4 | \(\Z/9\Z\) | not in database |
| $18$ | 18.6.1647129056757192000000000.2 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | - | - | 2 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| $\mu$-invariant(s) | - | - | - | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.