Properties

Label 825a1
Conductor $825$
Discriminant $-81675$
j-invariant \( -\frac{56197120}{3267} \)
CM no
Rank $1$
Torsion structure trivial

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Minimal Weierstrass equation

Minimal Weierstrass equation

Simplified equation

\(y^2+y=x^3-x^2-23x+53\) Copy content Toggle raw display (homogenize, simplify)
\(y^2z+yz^2=x^3-x^2z-23xz^2+53z^3\) Copy content Toggle raw display (dehomogenize, simplify)
\(y^2=x^3-30240x+2123280\) Copy content Toggle raw display (homogenize, minimize)

sage: E = EllipticCurve([0, -1, 1, -23, 53])
 
gp: E = ellinit([0, -1, 1, -23, 53])
 
magma: E := EllipticCurve([0, -1, 1, -23, 53]);
 
sage: E.short_weierstrass_model()
 
magma: WeierstrassModel(E);
 

Mordell-Weil group structure

\(\Z\)

Infinite order Mordell-Weil generator and height

sage: E.gens()
 
magma: Generators(E);
 

$P$ =  \(\left(1, 5\right)\) Copy content Toggle raw display
$\hat{h}(P)$ ≈  $0.26728146468235865198494989376$

Integral points

sage: E.integral_points()
 
magma: IntegralPoints(E);
 

\( \left(1, 5\right) \), \( \left(1, -6\right) \), \( \left(3, 1\right) \), \( \left(3, -2\right) \) Copy content Toggle raw display

Invariants

sage: E.conductor().factor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor: \( 825 \)  =  $3 \cdot 5^{2} \cdot 11$
sage: E.discriminant().factor()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant: $-81675 $  =  $-1 \cdot 3^{3} \cdot 5^{2} \cdot 11^{2} $
sage: E.j_invariant().factor()
 
gp: E.j
 
magma: jInvariant(E);
 
j-invariant: \( -\frac{56197120}{3267} \)  =  $-1 \cdot 2^{15} \cdot 3^{-3} \cdot 5 \cdot 7^{3} \cdot 11^{-2}$
Endomorphism ring: $\Z$
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
Sato-Tate group: $\mathrm{SU}(2)$
Faltings height: $-0.30121080929250401904572205629\dots$
Stable Faltings height: $-0.56945046136485408147918194516\dots$

BSD invariants

sage: E.rank()
 
magma: Rank(E);
 
Analytic rank: $1$
sage: E.regulator()
 
magma: Regulator(E);
 
Regulator: $0.26728146468235865198494989376\dots$
sage: E.period_lattice().omega()
 
gp: if(E.disc>0,2,1)*E.omega[1]
 
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
 
Real period: $3.3747765339753457527067305386\dots$
sage: E.tamagawa_numbers()
 
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
 
magma: TamagawaNumbers(E);
 
Tamagawa product: $ 2 $  = $ 1\cdot1\cdot2 $
sage: E.torsion_order()
 
gp: elltors(E)[1]
 
magma: Order(TorsionSubgroup(E));
 
Torsion order: $1$
sage: E.sha().an_numerical()
 
magma: MordellWeilShaInformation(E);
 
Analytic order of Ш: $1$ (exact)
sage: r = E.rank();
 
sage: E.lseries().dokchitser().derivative(1,r)/r.factorial()
 
gp: ar = ellanalyticrank(E);
 
gp: ar[2]/factorial(ar[1])
 
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
 
Special value: $ L'(E,1) $ ≈ $ 1.8040304299531682383269905694 $

Modular invariants

Modular form   825.2.a.b

sage: E.q_eigenform(20)
 
gp: xy = elltaniyama(E);
 
gp: x*deriv(xy[1])/(2*xy[2]+E.a1*xy[1]+E.a3)
 
magma: ModularForm(E);
 

\( q - q^{3} - 2 q^{4} + q^{7} + q^{9} - q^{11} + 2 q^{12} + q^{13} + 4 q^{16} + 6 q^{17} - 7 q^{19} + O(q^{20}) \) Copy content Toggle raw display

For more coefficients, see the Downloads section to the right.

sage: E.modular_degree()
 
magma: ModularDegree(E);
 
Modular degree: 72
$ \Gamma_0(N) $-optimal: yes
Manin constant: 1

Local data

This elliptic curve is not semistable. There are 3 primes of bad reduction:

sage: E.local_data()
 
gp: ellglobalred(E)[5]
 
magma: [LocalInformation(E,p) : p in BadPrimes(E)];
 
prime Tamagawa number Kodaira symbol Reduction type Root number ord($N$) ord($\Delta$) ord$(j)_{-}$
$3$ $1$ $I_{3}$ Non-split multiplicative 1 1 3 3
$5$ $1$ $II$ Additive 1 2 2 0
$11$ $2$ $I_{2}$ Non-split multiplicative 1 1 2 2

Galois representations

sage: rho = E.galois_representation();
 
sage: [rho.image_type(p) for p in rho.non_surjective()]
 
magma: [GaloisRepresentation(E,p): p in PrimesUpTo(20)];
 

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
$3$ 3B 3.4.0.1
sage: gens = [[6, 1, 13, 24], [1, 0, 6, 1], [4, 3, 9, 7], [25, 6, 24, 7], [3, 4, 8, 11], [1, 6, 0, 1], [21, 2, 10, 7]]
 
sage: GL(2,Integers(30)).subgroup(gens)
 
magma: Gens := [[6, 1, 13, 24], [1, 0, 6, 1], [4, 3, 9, 7], [25, 6, 24, 7], [3, 4, 8, 11], [1, 6, 0, 1], [21, 2, 10, 7]];
 
magma: sub<GL(2,Integers(30))|Gens>;
 

The image of the adelic Galois representation has level $30$, index $16$, genus $0$, and generators

$\left(\begin{array}{rr} 6 & 1 \\ 13 & 24 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 25 & 6 \\ 24 & 7 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 21 & 2 \\ 10 & 7 \end{array}\right)$

$p$-adic regulators

sage: [E.padic_regulator(p) for p in primes(5,20) if E.conductor().valuation(p)<2]
 

Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.

Iwasawa invariants

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Reduction type ss nonsplit add ord nonsplit ord ord ord ord ord ord ord ord ord ss
$\lambda$-invariant(s) 1,2 1 - 1 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 0,0

An entry - indicates that the invariants are not computed because the reduction is additive.

Isogenies

This curve has non-trivial cyclic isogenies of degree $d$ for $d=$ 3.
Its isogeny class 825a consists of 2 curves linked by isogenies of degree 3.

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
$2$ \(\Q(\sqrt{5}) \) \(\Z/3\Z\) Not in database
$3$ 3.1.300.1 \(\Z/2\Z\) Not in database
$6$ 6.0.270000.1 \(\Z/2\Z \oplus \Z/2\Z\) Not in database
$6$ 6.0.1235334375.1 \(\Z/3\Z\) Not in database
$6$ 6.2.450000.1 \(\Z/6\Z\) Not in database
$12$ deg 12 \(\Z/4\Z\) Not in database
$12$ deg 12 \(\Z/3\Z \oplus \Z/3\Z\) Not in database
$12$ 12.0.1822500000000.1 \(\Z/2\Z \oplus \Z/6\Z\) Not in database
$18$ 18.6.1346880880237797416008575439453125.3 \(\Z/9\Z\) Not in database
$18$ 18.0.7721710717374930375000000000000.2 \(\Z/6\Z\) 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.