Properties

Label 2.2.60.1-15.1-d7
Base field \(\Q(\sqrt{15}) \)
Conductor norm \( 15 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{15}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 15 \); class number \(2\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-15, 0, 1]))
 
gp: K = nfinit(Polrev([-15, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-15, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}-a{x}^{2}+\left(-4323a-16744\right){x}-307168a-1189678\)
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([1,1]),K([-16744,-4323]),K([-1189678,-307168])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([1,1]),Polrev([-16744,-4323]),Polrev([-1189678,-307168])], K);
 
magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,1],K![-16744,-4323],K![-1189678,-307168]]);
 

This is not a global minimal model: it is minimal at all primes except \((2,a+1)\). No global minimal model exists.

sage: E.is_global_minimal_model()
 

Mordell-Weil group structure

\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

Mordell-Weil generators

$P$$\hat{h}(P)$Order
$\left(108 a + 430 : -3388 a - 13054 : 1\right)$$2.6006256879862522510964662859669826361$$\infty$
$\left(-\frac{27}{2} a - 56 : \frac{137}{4} a + \frac{515}{4} : 1\right)$$0$$2$
$\left(-13 a - 54 : 33 a + 124 : 1\right)$$0$$2$

Invariants

Conductor: $\frak{N}$ = \((a)\) = \((3,a)\cdot(5,a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: $N(\frak{N})$ = \( 15 \) = \(3\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: $\Delta$ = $10497600$
Discriminant ideal: $(\Delta)$ = \((10497600)\) = \((2,a+1)^{12}\cdot(3,a)^{16}\cdot(5,a)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: $N(\Delta)$ = \( 110199605760000 \) = \(2^{12}\cdot3^{16}\cdot5^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
Minimal discriminant: $\frak{D}_{\mathrm{min}}$ = \((164025)\) = \((3,a)^{16}\cdot(5,a)^{4}\)
Minimal discriminant norm: $N(\frak{D}_{\mathrm{min}})$ = \( 26904200625 \) = \(3^{16}\cdot5^{4}\)
j-invariant: $j$ = \( \frac{272223782641}{164025} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: $\mathrm{End}(E)$ = \(\Z\)   
Geometric endomorphism ring: $\mathrm{End}(E_{\overline{\Q}})$ = \(\Z\)    (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{ST}(E)$ = $\mathrm{SU}(2)$

BSD invariants

Analytic rank: $r_{\mathrm{an}}$= \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: $r$ = \(1\)
Regulator: $\mathrm{Reg}(E/K)$ \( 2.6006256879862522510964662859669826361 \)
Néron-Tate Regulator: $\mathrm{Reg}_{\mathrm{NT}}(E/K)$ \( 5.2012513759725045021929325719339652722 \)
Global period: $\Omega(E/K)$ \( 1.9616888821913405749936556941151444563 \)
Tamagawa product: $\prod_{\frak{p}}c_{\frak{p}}$= \( 32 \)  =  \(1\cdot2^{4}\cdot2\)
Torsion order: $\#E(K)_{\mathrm{tor}}$= \(4\)
Special value: $L^{(r)}(E/K,1)/r!$ \( 2.6344644646404431079732199373426367678 \)
Analytic order of Ш: Ш${}_{\mathrm{an}}$= \( 1 \) (rounded)

BSD formula

$\displaystyle 2.634464465 \approx L'(E/K,1) \overset{?}{=} \frac{ \# Ш(E/K) \cdot \Omega(E/K) \cdot \mathrm{Reg}_{\mathrm{NT}}(E/K) \cdot \prod_{\mathfrak{p}} c_{\mathfrak{p}} } { \#E(K)_{\mathrm{tor}}^2 \cdot \left|d_K\right|^{1/2} } \approx \frac{ 1 \cdot 1.961689 \cdot 5.201251 \cdot 32 } { {4^2 \cdot 7.745967} } \approx 2.634464465$

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 

This elliptic curve is semistable. There are 2 primes $\frak{p}$ of bad reduction. Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.

$\mathfrak{p}$ $N(\mathfrak{p})$ Tamagawa number Kodaira symbol Reduction type Root number \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{N}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathfrak{D}_{\mathrm{min}}\)) \(\mathrm{ord}_{\mathfrak{p}}(\mathrm{den}(j))\)
\((2,a+1)\) \(2\) \(1\) \(I_0\) Good \(1\) \(0\) \(0\) \(0\)
\((3,a)\) \(3\) \(16\) \(I_{16}\) Split multiplicative \(-1\) \(1\) \(16\) \(16\)
\((5,a)\) \(5\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 15.1-d consists of curves linked by isogenies of degrees dividing 32.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 75.b2
\(\Q\) 720.c2