Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+y=x^3+2058x+72630\)
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(homogenize, simplify) |
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\(y^2z+yz^2=x^3+2058xz^2+72630z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+32928x+4648336\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(98, 1102\right) \) | $0.40534733922669702545458894916$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \([98:1102:1]\) | $0.40534733922669702545458894916$ | $\infty$ |
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| \( \left(392, 8820\right) \) | $0.40534733922669702545458894916$ | $\infty$ |
Integral points
\( \left(0, 269\right) \), \( \left(0, -270\right) \), \( \left(98, 1102\right) \), \( \left(98, -1103\right) \)
\([0:269:1]\), \([0:-270:1]\), \([98:1102:1]\), \([98:-1103:1]\)
\((0,\pm 2156)\), \((392,\pm 8820)\)
Invariants
| Conductor: | $N$ | = | \( 2205 \) | = | $3^{2} \cdot 5 \cdot 7^{2}$ |
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| Minimal Discriminant: | $\Delta$ | = | $-2836714452075$ | = | $-1 \cdot 3^{9} \cdot 5^{2} \cdot 7^{8} $ |
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| j-invariant: | $j$ | = | \( \frac{229376}{675} \) | = | $2^{15} \cdot 3^{-3} \cdot 5^{-2} \cdot 7$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.0728714150363499699946786957$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.77370816200124707910651241839$ |
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| $abc$ quality: | $Q$ | ≈ | $1.2666941259754438$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.664480013585716$ | |||
| Intrinsic torsion order: | $\#E(\mathbb Q)_\text{tors}^\text{is}$ | = | $1$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $0.40534733922669702545458894916$ |
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| Real period: | $\Omega$ | ≈ | $0.56681566115941669508946213301$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2\cdot2\cdot3 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $2.7570866409958876381512455261 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 2.757086641 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.566816 \cdot 0.405347 \cdot 12}{1^2} \\ & \approx 2.757086641\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 2688 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $3$ | $2$ | $I_{3}^{*}$ | additive | -1 | 2 | 9 | 3 |
| $5$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $7$ | $3$ | $IV^{*}$ | additive | 1 | 2 | 8 | 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 |
|---|---|---|---|
| $3$ | 3B.1.2 | 3.8.0.2 | $8$ |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has label 6.16.0-6.b.1.1, level \( 6 = 2 \cdot 3 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 0 & 5 \\ 5 & 0 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 3 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 2 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[6])$ is a degree-$18$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/6\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $3$ | additive | $2$ | \( 245 = 5 \cdot 7^{2} \) |
| $5$ | split multiplicative | $6$ | \( 441 = 3^{2} \cdot 7^{2} \) |
| $7$ | additive | $26$ | \( 45 = 3^{2} \cdot 5 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 2205.f
consists of 2 curves linked by isogenies of
degree 3.
Twists
The minimal quadratic twist of this elliptic curve is 735.d2, its twist by $21$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-3}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.588.1 | \(\Z/2\Z\) | not in database |
| $3$ | 3.1.3675.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.1037232.1 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.40516875.1 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.1.9529569000000.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $18$ | 18.0.30056466355253425451469188671875.1 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.272438055977283000000000000.2 | \(\Z/6\Z \oplus \Z/6\Z\) | 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 | ss | add | split | add | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 2,3 | - | 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
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.